Advanced Topology: From Point-Set to Algebraic
Advanced topology bridges the study of abstract spaces with deep algebraic invariants. From separation axioms and compactness to homotopy groups, homology, and manifold theory, these tools reveal the global shape of mathematical objects.
1. Point-Set Topology Review
Point-set topology (general topology) provides the axiomatic framework for continuity, convergence, and connectedness in abstract spaces. It generalizes metric space concepts to settings where no notion of distance is required.
Topological Spaces
A topological space is a pair (X, T) where X is a set and T is a topology on X. A topology T is a collection of subsets of X (called open sets) satisfying: the empty set and X are in T; any arbitrary union of sets in T is in T; and any finite intersection of sets in T is in T.
Topology Axioms
- empty set in T and X in T
- U_alpha in T for all alpha implies union of U_alpha in T
- U_1, ..., U_n in T implies U_1 intersect ... intersect U_n in T
Open Sets, Closed Sets, and Neighborhoods
A set C is closed if its complement X minus C is open. Closed sets are stable under arbitrary intersections and finite unions. A neighborhood of a point x is any open set containing x. The closure of a set A (written A-bar) is the smallest closed set containing A, equal to the intersection of all closed sets containing A. The interior of A is the largest open set contained in A.
The boundary of A is the set of points in the closure of A that are not in the interior of A. A point x is a limit point (or accumulation point) of A if every neighborhood of x contains a point of A other than x itself. The derived set A' consists of all limit points of A, and A-bar equals A union A'.
Bases and Subbases
A basis B for a topology T is a collection of open sets such that every open set is a union of basis elements. A collection B is a basis if and only if: for each x in X, some B in B contains x; and if x is in B_1 intersect B_2 for B_1, B_2 in B, then some B_3 in B contains x and is contained in B_1 intersect B_2.
A subbasis S for T is a collection whose finite intersections form a basis. The topology generated by S consists of all unions of finite intersections of elements of S. The standard topology on R has basis consisting of open intervals (a, b).
Product Topology
Given spaces X and Y, the product topology on X cross Y has as basis the collection of sets U cross V where U is open in X and V is open in Y. For infinite products, the product topology (Tychonoff topology) has as subbasis the preimages of open sets under each projection map.
Subspace Topology
Given a topological space (X, T) and a subset A of X, the subspace topology on A is the collection of sets of the form U intersect A where U is in T. This makes A into a topological space and the inclusion map into X continuous. For example, [0,1] inherits its topology from R, and [0, 1/2) is open in the subspace [0,1] even though it is not open in R.
Continuous Maps and Homeomorphisms
A map f from X to Y is continuous if and only if the preimage of every open set in Y is open in X. Equivalently, f is continuous at x if for every neighborhood V of f(x), the preimage of V is a neighborhood of x. A homeomorphism is a continuous bijection with a continuous inverse. Homeomorphic spaces are topologically identical — topology studies properties invariant under homeomorphism.
2. Compactness
Compactness is one of the central concepts in topology, capturing the idea that a space is in some sense "finite" or "small enough" to behave like a closed bounded subset of Euclidean space.
Definition and Open Cover Characterization
A topological space X is compact if every open cover of X has a finite subcover. That is, if X is contained in the union of a collection of open sets, then finitely many of those open sets already cover X. This definition avoids any reference to metrics or coordinates.
Heine-Borel Theorem
In R^n with the standard topology, a subset is compact if and only if it is closed and bounded. This elegant characterization is specific to Euclidean space and fails in general metric spaces. In an infinite-dimensional Banach space, the closed unit ball is closed and bounded but not compact.
Key Consequences of Compactness
- A continuous image of a compact space is compact
- A continuous real-valued function on a compact space attains its maximum and minimum
- A continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- Compact subsets of Hausdorff spaces are closed
Sequential Compactness and Limit Point Compactness
A space is sequentially compact if every sequence has a convergent subsequence. A space is limit point compact (or Bolzano-Weierstrass compact) if every infinite subset has a limit point. In metric spaces, all three notions of compactness coincide. In general topological spaces, they can differ.
The sequence [0,1] is sequentially compact: the Bolzano-Weierstrass theorem guarantees that every bounded real sequence has a convergent subsequence. This equivalence in metric spaces is a profound connection between analysis and topology.
Tychonoff's Theorem
Tychonoff's theorem states that an arbitrary product of compact spaces is compact in the product topology. This is one of the most important theorems in point-set topology. Its proof requires the Axiom of Choice (indeed, it is equivalent to the Axiom of Choice), typically using either Zorn's lemma or the ultrafilter lemma.
Application: Stone-Cech Compactification
Tychonoff's theorem implies that the Stone-Cech compactification of any completely regular space exists. For a space X, the Stone-Cech compactification beta(X) is the largest compactification of X in the sense that every bounded continuous function on X extends to beta(X).
3. Connectedness
Connectedness captures the intuitive idea that a space is "in one piece." Like compactness, it is a topological invariant preserved by continuous maps.
Connected Spaces
A topological space X is connected if it cannot be written as the disjoint union of two nonempty open sets. Equivalently, X is connected if the only subsets that are both open and closed (clopen) are the empty set and X itself. The continuous image of a connected space is connected.
Intermediate Value Theorem (Topological Form)
If f is a continuous map from a connected space X to R, and a and b are values with a less than b, then f takes every value between a and b. The classical IVT follows because closed intervals [a,b] in R are connected.
Path-Connectedness
A space X is path-connected if for any two points x and y in X, there exists a continuous path gamma from [0,1] to X with gamma(0) = x and gamma(1) = y. Path-connectedness implies connectedness, but the converse fails. The classic counterexample is the topologist's sine curve: the closure of the graph of sin(1/x) for x greater than 0, which is connected but not path-connected.
Components and Local Connectedness
The connected components of a space are the maximal connected subsets. They partition X into disjoint closed sets. The path-components are the maximal path-connected subsets. A space is locally connected if every neighborhood of every point contains a connected open neighborhood. Locally connected spaces have the property that connected components are open.
The Cantor set is an example of a totally disconnected compact space: every connected component is a single point. Despite this, the Cantor set is uncountable and in some sense quite large.
4. Separation Axioms
Separation axioms classify topological spaces by how well their open sets can distinguish between points and sets. They form a hierarchy from T0 (the weakest) to T4 (the strongest commonly used).
T0 (Kolmogorov)
For any two distinct points x and y, there exists an open set containing one but not the other. The Sierpinski space (two points, one open) is T0 but not T1.
T1 (Frechet)
For any two distinct points x and y, there exist open sets U containing x but not y, and V containing y but not x. In T1 spaces, all singleton sets are closed.
T2 (Hausdorff)
Any two distinct points have disjoint open neighborhoods. Hausdorff spaces are the most commonly assumed in analysis: limits of sequences are unique, and compact subsets are closed.
T3 (Regular)
A T1 space where any point and disjoint closed set have disjoint open neighborhoods. Equivalently, every neighborhood of a point contains a closed neighborhood.
T4 (Normal)
A T1 space where any two disjoint closed sets have disjoint open neighborhoods. All metric spaces are normal (T4). All compact Hausdorff spaces are normal. Normal spaces are the natural setting for Urysohn's lemma and the Tietze extension theorem.
Urysohn's Lemma
Urysohn's lemma is one of the most powerful tools in point-set topology. It states: a topological space X is normal if and only if for any two disjoint closed sets A and B, there exists a continuous function f from X to [0,1] with f equal to 0 on A and f equal to 1 on B. The "if" direction is easy; the "only if" direction requires a careful dyadic rational construction.
The proof constructs open sets U(r) for each dyadic rational r in [0,1] such that A is contained in U(r), the closure of U(r) is contained in U(s) for r less than s, and the complement of B equals the union of all U(r). The function f(x) is then defined as the infimum of all r such that x is in U(r).
Tietze Extension Theorem
The Tietze extension theorem states that in a normal space X, any continuous function from a closed subset A to R (or to [a,b]) can be extended to a continuous function on all of X. This theorem, proved using Urysohn's lemma iteratively, is fundamental in algebraic topology when constructing homotopies on CW complexes.
5. Metric Spaces and Completeness
Metric spaces occupy the intersection of topology and analysis. They carry enough structure to define Cauchy sequences and completeness, which are essential for many existence theorems.
Complete Metric Spaces
A metric space (X, d) is complete if every Cauchy sequence converges to a point in X. A sequence (x_n) is Cauchy if for every epsilon greater than 0, there exists N such that for all m, n greater than N, d(x_m, x_n) is less than epsilon. The real numbers and R^n are complete. The rationals Q are not complete: the sequence of rational approximations to sqrt(2) is Cauchy but does not converge in Q.
Baire Category Theorem
A subset of a topological space is nowhere dense if its closure has empty interior. A set is meager (first category) if it is a countable union of nowhere-dense sets. The Baire Category Theorem states that a complete metric space is not meager in itself: equivalently, the intersection of countably many dense open sets is dense.
Consequences of BCT
- There exist continuous functions on [0,1] that are nowhere differentiable
- The set of differentiable functions is meager in the space of continuous functions
- The principle of uniform boundedness in Banach space theory
- The open mapping theorem and closed graph theorem in functional analysis
Completion of a Metric Space
Every metric space X has a completion X-hat: a complete metric space containing X as a dense subspace. The completion is unique up to isometry. It is constructed as equivalence classes of Cauchy sequences in X, where two sequences are equivalent if their interleaving is Cauchy with limit 0. The real numbers are the completion of the rationals under the standard metric.
Cantor's Intersection Theorem
In a complete metric space, if (F_n) is a decreasing sequence of nonempty closed sets with diameters converging to 0, then the intersection of all F_n is exactly one point. This theorem characterizes completeness: a metric space is complete if and only if Cantor's intersection theorem holds. It is used to prove the contraction mapping theorem (Banach fixed point theorem), a cornerstone of existence theory for differential equations.
6. Homotopy Theory
Homotopy theory studies spaces and maps up to continuous deformation. Two maps are homotopic if one can be continuously deformed into the other. This gives rise to algebraic invariants that distinguish topological spaces.
Homotopy of Maps and Paths
A homotopy between maps f and g from X to Y is a continuous map H from X cross [0,1] to Y with H(x,0) = f(x) and H(x,1) = g(x) for all x. If f and g agree on a subspace A, the homotopy is relative to A if H(a,t) = f(a) for all a in A and all t. Paths are continuous maps gamma from [0,1] to X. Two paths with the same endpoints are path-homotopic if there is a homotopy between them relative to the endpoints.
The Fundamental Group
Given a pointed space (X, x_0), the fundamental group pi_1(X, x_0) consists of path-homotopy classes of loops based at x_0 (paths starting and ending at x_0). The group operation is concatenation: given loops alpha and beta, their product is the loop that first traverses alpha then traverses beta (each at double speed). The identity element is the constant loop at x_0, and the inverse of a loop is the loop traversed in reverse.
Well-Definedness
The fundamental group is well-defined because path-homotopy classes are closed under concatenation: if alpha is homotopic to alpha' and beta is homotopic to beta' (all relative to endpoints), then the concatenation alpha*beta is homotopic to alpha'*beta'. The group axioms (associativity up to homotopy, identity, inverses) are all verified via explicit homotopy constructions.
Van Kampen's Theorem
Van Kampen's theorem computes the fundamental group of a space as a pushout of fundamental groups of simpler pieces. If X is the union of path-connected open sets U and V, both containing the basepoint x_0, and U intersect V is path-connected, then pi_1(X, x_0) is the free product of pi_1(U) and pi_1(V) amalgamated over pi_1(U intersect V).
In categorical terms, pi_1(X) is the pushout in the category of groups of the diagram pi_1(U intersect V) to pi_1(U) and to pi_1(V). The amalgamated free product is the free product with the relation that the image of any element of pi_1(U intersect V) in pi_1(U) equals its image in pi_1(V).
Covering Spaces
A covering space of X is a space X-tilde together with a continuous surjective map p from X-tilde to X such that every point of X has an evenly covered open neighborhood. The universal cover of X (assuming X is path-connected, locally path-connected, and semi-locally simply connected) is the unique simply connected covering space. The fundamental group pi_1(X, x_0) acts on the fiber p-inverse of x_0 by deck transformations.
Classification of Covering Spaces
There is a correspondence between connected covering spaces of X and subgroups of pi_1(X, x_0). The universal cover corresponds to the trivial subgroup. A regular (normal) covering corresponds to a normal subgroup H, and the deck transformation group is isomorphic to pi_1(X)/H.
7. Fundamental Group Computations
Computing fundamental groups requires combining van Kampen's theorem, knowledge of simple spaces, and geometric insight into how spaces are assembled.
Circle S^1
pi_1(S^1) is isomorphic to Z. A generator is the loop that winds once counterclockwise. The integer n corresponds to the homotopy class of a loop winding n times. The universal cover is R with covering map t mapping to e^(2 pi i t).
Torus T^2
pi_1(T^2) is isomorphic to Z cross Z. The two generators correspond to the longitudinal and meridional loops. The torus is S^1 cross S^1, and pi_1 of a product is the product of the fundamental groups (when each factor is path-connected).
RP^2
The real projective plane has pi_1(RP^2) isomorphic to Z/2Z. This follows from the double cover S^2 over RP^2: the deck transformation group is Z/2Z acting by the antipodal map. The generator is the path connecting antipodal points.
Bouquet of Circles
A bouquet of n circles (wedge sum of n copies of S^1) has fundamental group equal to the free group F_n on n generators. By van Kampen's theorem applied iteratively, each circle contributes a free generator with no relations.
Surfaces and their Fundamental Groups
A compact orientable surface of genus g has fundamental group with presentation: generators a_1, b_1, ..., a_g, b_g and a single relation [a_1, b_1][a_2, b_2]...[a_g, b_g] = 1, where [a,b] denotes the commutator aba^(-1)b^(-1). For genus 0 (sphere), the group is trivial. For genus 1 (torus), the presentation simplifies to aba^(-1)b^(-1) = 1, giving the abelian group Z cross Z.
Non-orientable surfaces such as the Klein bottle have fundamental group with presentation: generators a, b and relation abab^(-1) = 1. The Klein bottle cannot be embedded in R^3 without self-intersection. The projective plane has pi_1 = Z/2Z as noted above.
8. Singular Homology
Singular homology assigns to each topological space a sequence of abelian groups H_0, H_1, H_2, ... that measure the presence of holes of different dimensions. It is functorial: continuous maps induce group homomorphisms, and homotopic maps induce the same homomorphism.
Chain Complexes and Boundary Maps
The n-th singular chain group C_n(X) is the free abelian group on singular n-simplices (continuous maps from the standard n-simplex delta^n to X). The boundary map partial_n from C_n(X) to C_(n-1)(X) is defined by alternating face maps: for a singular n-simplex sigma, partial_n(sigma) is the alternating sum of the (n-1)-dimensional faces of sigma.
Fundamental Identity
partial_(n-1) composed with partial_n = 0
The boundary of a boundary is zero. This means the image of partial_(n+1) is contained in the kernel of partial_n. We define n-cycles Z_n = ker(partial_n), n-boundaries B_n = im(partial_(n+1)), and the n-th homology group H_n(X) = Z_n / B_n.
Exact Sequences
A sequence of abelian groups and homomorphisms ... to A to B to C to ... is exact at B if the image of the incoming map equals the kernel of the outgoing map. Short exact sequences 0 to A to B to C to 0 generalize the splitting of a group as a product. Long exact sequences connect homology groups across dimensions and are the primary computational tool in algebraic topology.
Mayer-Vietoris Sequence
The Mayer-Vietoris sequence is the homological analogue of van Kampen's theorem. If X is the union of open sets A and B, then there is a long exact sequence:
... to H_n(A intersect B) to H_n(A) direct-sum H_n(B) to H_n(X) to H_(n-1)(A intersect B) to ...
The connecting homomorphism and the maps in this sequence are induced by inclusions. The sequence terminates at H_0 and can be used to compute the homology of spheres, tori, and any space decomposable into simpler pieces.
Homology of Spheres
For the n-sphere S^n, the homology groups are: H_0(S^n) = Z (one connected component), H_n(S^n) = Z (the fundamental cycle), and H_k(S^n) = 0 for all other k. This is computed by induction using Mayer-Vietoris, decomposing S^n as the union of two hemispheres, each homeomorphic to a disk D^n (which is contractible), with intersection homotopy equivalent to S^(n-1).
9. Cohomology
Cohomology is the dual theory to homology. While homology counts holes detected by cycles, cohomology uses cochains (functions on chains) and carries a multiplicative ring structure: the cup product.
Cochain Complexes and Cohomology Groups
The n-th cochain group C^n(X; G) is the group Hom(C_n(X), G) of homomorphisms from singular chains to an abelian group G (often Z or a field). The coboundary map delta^n from C^n to C^(n+1) is the adjoint of the boundary map. Cocycles Z^n = ker(delta^n) and coboundaries B^n = im(delta^(n-1)) give cohomology H^n(X; G) = Z^n / B^n.
Cup Product
The cup product is a bilinear map from H^p(X; R) cross H^q(X; R) to H^(p+q)(X; R) for a ring R. It makes the direct sum of all cohomology groups into a graded ring called the cohomology ring of X. The cup product is graded-commutative: alpha cup beta equals (-1)^(pq) times (beta cup alpha) for classes of degrees p and q.
Cohomology Rings
- H*(S^n; Z) = Z[x] / (x^2) where x has degree n
- H*(CP^n; Z) = Z[x] / (x^(n+1)) where x has degree 2
- H*(T^n; Z) = exterior algebra on n generators of degree 1
- H*(RP^n; Z/2Z) = (Z/2Z)[x] / (x^(n+1)) where x has degree 1
de Rham Cohomology
For smooth manifolds, de Rham cohomology H^k_dR(M) is defined using differential forms. The de Rham complex consists of the spaces of k-forms connected by the exterior derivative d. Since d composed with d equals 0, we have closed k-forms (ker d) and exact k-forms (im d from (k-1)-forms). The k-th de Rham cohomology is the quotient of closed forms by exact forms.
The de Rham theorem states that de Rham cohomology with real coefficients is naturally isomorphic to singular cohomology with real coefficients. This bridges differential geometry and algebraic topology.
Universal Coefficient Theorem
The universal coefficient theorem relates cohomology with coefficients in G to homology. There is a (non-naturally split) short exact sequence: 0 to Ext^1(H_(n-1)(X), G) to H^n(X; G) to Hom(H_n(X), G) to 0. For G a field, the Ext term vanishes and cohomology is the vector space dual of homology. This explains why cohomology often carries more information than homology when integer coefficients are used.
10. CW Complexes
CW complexes (closure-finite, weak topology) are a class of topological spaces built by attaching cells of increasing dimension. They provide a flexible and computable framework for algebraic topology.
Cell Attachment
A CW complex is built inductively. Start with a discrete set of points (the 0-skeleton X^0). Attach n-cells e^n (open n-disks) to the (n-1)-skeleton via attaching maps phi from the boundary sphere S^(n-1) to X^(n-1). The n-skeleton X^n is the pushout of X^(n-1) and the disjoint union of n-disks along the attaching maps. The full complex X has the weak topology: a set is closed in X if and only if its intersection with each closed cell is closed.
Examples of CW Structures
- S^n: one 0-cell and one n-cell attached by the unique map from S^(n-1) to a point
- T^2: one 0-cell, two 1-cells, one 2-cell attached via the word aba^(-1)b^(-1)
- RP^n: one cell in each dimension 0 through n
- CP^n: one cell in each even dimension 0, 2, 4, ..., 2n
Cellular Homology
Cellular homology is a powerful computational tool for CW complexes. The cellular chain group C_n^cell(X) is the free abelian group on the n-cells of X. The cellular boundary map is computed using the degrees of the attaching maps. The resulting chain complex, though far smaller than the singular chain complex, computes the same homology groups.
The degree of a map f from S^n to S^n is the integer d such that f induces multiplication by d on H_n(S^n) = Z. Degree can be computed as an algebraic count of preimages for regular values, with sign determined by whether f preserves or reverses orientation near each preimage.
Euler Characteristic
The Euler characteristic chi(X) is the alternating sum of Betti numbers: chi(X) = sum of (-1)^n rank(H_n(X)). For a CW complex, chi(X) equals the alternating sum of the number of n-cells: chi(X) = (number of 0-cells) - (number of 1-cells) + (number of 2-cells) - ... This is a topological invariant independent of the CW structure. For a compact surface of genus g, chi = 2 - 2g.
11. Manifolds
Manifolds are topological spaces that locally look like Euclidean space. They are the primary objects of study in differential geometry and much of modern physics.
Topological Manifolds
A topological n-manifold (possibly with boundary) is a Hausdorff second-countable topological space in which every point has a neighborhood homeomorphic to R^n (or to the half-space R^(n-1) cross [0, infinity) for boundary points). Manifolds without boundary are sometimes called closed manifolds when compact. Examples include: curves (1-manifolds), surfaces (2-manifolds), and all smooth manifolds of any dimension.
Smooth Manifolds: Charts and Atlases
A smooth (differentiable) manifold is a topological manifold equipped with a smooth atlas: a collection of charts (homeomorphisms from open subsets of M to open subsets of R^n) such that all transition maps (compositions of charts) are smooth. Two atlases define the same smooth structure if their union is also a smooth atlas.
Exotic Structures
Milnor's 1956 result showed that S^7 admits exotic smooth structures (smooth manifolds homeomorphic but not diffeomorphic to the standard S^7). In dimension 4, there are uncountably many exotic smooth structures on R^4, a phenomenon unique to dimension 4.
Orientability
An n-manifold M is orientable if there exists an atlas where all transition maps have positive Jacobian determinant. Equivalently, M is orientable if H_n(M; Z) is isomorphic to Z (for compact connected M). Non-orientable surfaces include the Mobius band and the Klein bottle. The Mobius band has H_1 = Z but its boundary is connected (unlike an annulus whose boundary has two components).
Classification of Compact Surfaces
Every compact connected 2-manifold without boundary is homeomorphic to exactly one of: the sphere S^2; a connected sum of g tori T^2 # T^2 # ... (g copies) for g at least 1 (orientable surfaces of genus g); or a connected sum of k real projective planes RP^2 # RP^2 # ... (k copies) for k at least 1 (non-orientable surfaces). The Euler characteristic and orientability together classify surfaces completely.
12. Fixed Point Theorems
Fixed point theorems guarantee that certain maps must fix some point. They have profound applications in analysis, game theory, and economics.
Brouwer Fixed Point Theorem
Every continuous map f from the closed n-disk D^n to itself has at least one fixed point. The proof using homology: if f had no fixed point, we could construct a retraction r from D^n to S^(n-1) (the boundary sphere) by shooting a ray from f(x) through x to the boundary. But S^(n-1) is not a retract of D^n because H_(n-1)(S^(n-1)) = Z while H_(n-1)(D^n) = 0, and a retraction would force the inclusion-induced map on homology to be injective and the retraction-induced map to be its left inverse — a contradiction.
Lefschetz Fixed Point Theorem
The Lefschetz fixed point theorem generalizes Brouwer's to arbitrary compact spaces. Given a continuous map f from a compact polyhedron X to itself, define the Lefschetz number L(f) as the alternating sum over n of the traces of the maps induced by f on H_n(X; Q). If L(f) is not zero, then f has at least one fixed point. For the identity map, L(id) = chi(X). If chi(X) is not zero, then every continuous self-map of X has a fixed point — for example, every self-map of CP^n has a fixed point.
Computation Example
For the torus T^2, chi(T^2) = 0. Indeed, the translation map (which shifts the torus by a fixed vector) has no fixed points. The Lefschetz number of any such translation is 0, consistent with the theorem (L(f) = 0 does not guarantee fixed points, but L(f) not 0 guarantees one exists).
Borsuk-Ulam Theorem
The Borsuk-Ulam theorem states that for every continuous map f from S^n to R^n, there exists a point x in S^n such that f(x) = f(-x). Equivalently, no continuous map from S^n to S^(n-1) satisfies f(-x) = -f(x) (antipode-preserving). The case n = 1 states that every continuous function on the circle that is antipodal-equivariant must have a zero — a consequence of the intermediate value theorem. For n = 2, it implies that at any moment, there exist two antipodal points on Earth with the same temperature and pressure.
The proof uses mod-2 cohomology or the degree of maps. A corollary is the ham sandwich theorem: given n measurable subsets of R^n, there exists a hyperplane that simultaneously bisects all of them.
13. Practice Problems with Solutions
Work through these problems to solidify your understanding of advanced topology. Click each problem to reveal the solution.
Problem 1: Show that the product of two compact spaces is compact (finite case of Tychonoff).
Solution
Let X and Y be compact, and let U be an open cover of X cross Y. For each x in X, the slice x cross Y is homeomorphic to Y and hence compact. Cover x cross Y with finitely many sets from U, say U_1, ..., U_k. For each U_i, write it as a union of basic open sets of the form V cross W. Intersect the finitely many V components corresponding to the chosen cover to get an open neighborhood N(x) of x.
The sets N(x) as x varies cover X. Since X is compact, finitely many N(x_1), ..., N(x_m) suffice. For each x_j, we chose finitely many U-sets covering N(x_j) cross Y. The union of all these finitely many U-sets (across all j from 1 to m) covers all of X cross Y. This gives a finite subcover, so X cross Y is compact.
The key idea is the tube lemma: if x cross Y is covered by finitely many open sets, there is a tube N(x) cross Y (for some neighborhood N(x) of x) contained in their union.
Problem 2: Compute pi_1 of the figure eight (wedge of two circles).
Solution
Let X = S^1 wedge S^1, the wedge point being p. Write X as the union of open sets U and V, where U is a small open neighborhood of the first circle (thickened slightly past the wedge point into the second circle) and V is similarly defined for the second circle.
Then U deformation retracts to S^1, so pi_1(U) = Z. Similarly pi_1(V) = Z. The intersection U intersect V deformation retracts to the wedge point, so pi_1(U intersect V) = 1 (trivial group). By van Kampen's theorem, pi_1(X) is the free product of Z and Z amalgamated over the trivial group, which is Z * Z, the free group on two generators.
Note: Z * Z is non-abelian. The generator a corresponds to a loop around the first circle, b to a loop around the second. The word aba^(-1)b^(-1) is nontrivial in the free group.
Problem 3: Compute H_*(S^2) using Mayer-Vietoris.
Solution
Decompose S^2 as the union of the open northern hemisphere U (slightly past the equator) and the open southern hemisphere V. Both U and V are contractible (homeomorphic to open disks), so H_n(U) = H_n(V) = 0 for n greater than 0, and H_0(U) = H_0(V) = Z. The intersection U intersect V is an open annular band around the equator, homotopy equivalent to S^1.
The Mayer-Vietoris sequence in degree 2: H_2(U intersect V) to H_2(U) direct-sum H_2(V) to H_2(S^2) to H_1(U intersect V) to H_1(U) direct-sum H_1(V) becomes: 0 to 0 to H_2(S^2) to Z to 0. So H_2(S^2) = Z.
In degree 1: H_1(U intersect V) to H_1(U) direct-sum H_1(V) to H_1(S^2) to H_0(U intersect V) to H_0(U) direct-sum H_0(V) becomes: Z to 0 to H_1(S^2) to Z to Z direct-sum Z. The map Z to Z direct-sum Z sends a generator of H_0(S^1) to (1, 1), which is injective, so H_1(S^2) = 0. And H_0(S^2) = Z since S^2 is connected.
Problem 4: Show every compact metric space is separable.
Solution
A topological space is separable if it has a countable dense subset. For a compact metric space (X, d), consider the open cover by balls of radius 1/n for each positive integer n. Compactness gives a finite subcover: finitely many balls of radius 1/n cover X. Let D_n be the finite set of their centers.
Let D be the countable union of all D_n. We claim D is dense in X. Given any point x in X and epsilon greater than 0, choose n with 1/n less than epsilon. Then x is in some ball of radius 1/n with center c in D_n, so d(x, c) is less than 1/n which is less than epsilon. Thus every ball around x contains a point of D, so D is dense. Since D is a countable union of finite sets, it is countable. Therefore X is separable.
Problem 5: Prove the Euler characteristic of a torus is 0 using CW structure.
Solution
Give T^2 the standard CW structure with one 0-cell (vertex v), two 1-cells (edges a and b), and one 2-cell (the square with boundary identified as the word aba^(-1)b^(-1)). Count: one 0-cell, two 1-cells, one 2-cell.
The Euler characteristic is chi = (number of 0-cells) - (number of 1-cells) + (number of 2-cells) = 1 - 2 + 1 = 0. Alternatively, using Betti numbers: H_0(T^2) = Z (rank 1), H_1(T^2) = Z direct-sum Z (rank 2), H_2(T^2) = Z (rank 1), so chi = 1 - 2 + 1 = 0.
This matches the formula chi = 2 - 2g for orientable surfaces of genus g: the torus has genus 1, giving chi = 2 - 2(1) = 0. A sphere (genus 0) has chi = 2, a double torus has chi = -2.
Problem 6: Show that S^1 is not simply connected using covering spaces.
Solution
A space is simply connected if it is path-connected and has trivial fundamental group. We want to show pi_1(S^1) is not trivial. Consider the covering map p from R to S^1 defined by p(t) = (cos(2 pi t), sin(2 pi t)).
The loop gamma(t) = (cos(2 pi t), sin(2 pi t)) in S^1 is a loop based at (1,0). Its unique lift starting at 0 in R is the path t maps to t, which starts at 0 and ends at 1. Since 0 and 1 are distinct points in R, the lift is not a loop in R, so gamma is not null-homotopic in S^1.
This argument uses the lifting criterion: a loop in S^1 is null-homotopic if and only if its lift to R is a loop (i.e., closes up). The integer winding number is the endpoint of the lift minus its starting point, and this defines the isomorphism pi_1(S^1) to Z.
Problem 7: Prove that RP^2 cannot be embedded in R^2 using homology.
Solution
Suppose RP^2 embeds in R^2. By the Jordan curve theorem generalization (Alexander duality), an embedded compact 2-manifold without boundary in R^2 would have to separate R^2 into components. But RP^2 is a closed 2-manifold (compact, without boundary) that is non-orientable.
A more direct homology argument: RP^2 has H_2(RP^2; Z) = 0 (since it is non-orientable), so it is not a closed orientable 2-manifold. On the other hand, any compact 2-manifold embedded in R^2 must bound a region, making it homeomorphic to a sphere or a disk. But RP^2 is neither: pi_1(RP^2) = Z/2Z is nontrivial.
A cleaner argument: RP^2 is a compact 2-manifold without boundary and every compact 2-manifold without boundary embeds in R^3 but those that are non-orientable (like RP^2) do not embed in R^2. The intersection form on H_1 detects this: a non-orientable surface has an element of H_1 whose self-intersection number is odd, which is impossible for a surface in R^2 (where all self-intersections are even by the plane's orientation).
Problem 8: Use the Baire Category Theorem to show the rationals are not a countable intersection of open sets in R.
Solution
Suppose for contradiction that Q equals the intersection of countably many open sets U_n in R. Each U_n is open and contains Q, so each U_n is dense in R (since Q is dense and U_n contains Q). By the Baire Category Theorem, R (being a complete metric space) has the property that the intersection of countably many dense open sets is dense.
However, Q is countable, so we can write Q as a sequence q_1, q_2, q_3, ... The complement R minus Q consists of the irrationals. For each rational q_n, the set R minus q_n (a single point removed) is open and dense. So R minus Q equals the intersection of (R minus q_n) over all n.
Now Q intersect (R minus Q) = empty set. But if Q is a G_delta set (intersection of open sets) and R minus Q is also a G_delta (as just argued), then R is the union of two meager sets (since both Q and R minus Q would have to be meager by a further BCT argument). But BCT says R is not meager, contradiction. Hence Q is not a G_delta set.
Quick Reference: Key Theorems
| Theorem | Statement | Key Hypothesis |
|---|---|---|
| Heine-Borel | Compact in R^n iff closed and bounded | Euclidean space R^n |
| Tychonoff | Product of compacts is compact | Axiom of Choice |
| Urysohn's Lemma | Disjoint closed sets separated by function | Normal (T4) space |
| Tietze Extension | Continuous functions extend from closed subsets | Normal space |
| Baire Category | Countable intersection of dense opens is dense | Complete metric space |
| Van Kampen | pi_1(U union V) = pi_1(U) *_(pi_1(U intersect V)) pi_1(V) | U, V, U intersect V path-connected open |
| Mayer-Vietoris | Long exact sequence for homology of union | X = int(A) union int(B) |
| Brouwer FPT | Every self-map of D^n has a fixed point | Continuous map, D^n compact convex |
| Lefschetz FPT | L(f) not 0 implies fixed point exists | Compact polyhedron |
| Borsuk-Ulam | f(x) = f(-x) for some x in S^n | f continuous from S^n to R^n |
| de Rham | H^k_dR(M; R) cong H^k_sing(M; R) | Smooth manifold M |
Recommended Study Path
Advanced topology draws on analysis, algebra, and geometric intuition. Follow this path for systematic mastery.
Foundation
- Point-set topology (Munkres Part I)
- Metric space theory and completeness
- Real analysis and measure theory
- Abstract algebra (groups, rings, modules)
Core Algebraic Topology
- Fundamental group and van Kampen
- Covering space theory
- Singular homology and Mayer-Vietoris
- CW complexes and cellular homology
- Cohomology and cup products
Advanced Topics
- Higher homotopy groups pi_n
- Fiber bundles and spectral sequences
- Characteristic classes
- K-theory and cobordism
- Surgery theory and Poincare duality
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