MAT 530 Topology
Fall 2004
Instructor: Vladlen
Timorin
e-mail: mailto:timorin@math.sunysb.edu
Office: 4-102 Phone: 2-8265
Office hours: WF 9-11 or by appointment
Textbook: Munkres Topology, 2nd edition, Prentice Hall 1999
Syllabus (from the Graduate Student Handbook):
Basic point set
topology
First I tried to illustrate the following folklore definition of topology: topology studies the properties of geometric objects that remain unchanged under all continuous deformations. We gave examples of several topological invariants (without any proofs). I made some remarks on the topological nature of the famous Euler's formula in combinatorics of convex polyhedra.
The next were the formal definitions of metric and topological spaces, bases and subbases in topological spaces (i.e., a description of different ways to define topology on a set). Of course, the most interesting examples of topological spaces are metric spaces. Examples of metric spaces: Euclidean spaces, max-norm on a finite-dimensional real vector space, p-adic metrics on the field of rational numbers, etc.
Some other examples of topological spaces: the 3 essentially different topologies on a 2-point set, the order topology of a linearly ordered set, we also know how to define topology on a partially ordered set such that any pair of elements admits a lower bound.
Closed sets are complemets to the open sets; they form the so called closed topology and provide an alternative way to define topological spaces. We defined the closure and the interior of any subset in a topological space, and studied some properties of the closure.
The notion of continuous maps is probably the most important in all topology. It generalizes continuous functions of real arguments. We gave a topological definition of continuity that does not appeal to a metric structure or any other additional structures and thus can be applied to any topological spaces. Homeomorphisms are bijective continuous maps with continuous inverses. Intuitively, the continuity prevents "cutting", and the continuity of the inverse map prevents "glueing". Homeomorphic topological spaces have the same topological properties.
A quotient space, i.e. the set of equivalence classes of some equivalence relation on a topological space, can be equipped with a natural topology - the quotient topology. This is the largest (finest, strongest) topology such that the canonical projection (from the space to the quotient-space) is continuous. The direct (Cartesian) product of two topological spaces also carries a natural direct product topology that is the smallest (coarsest, weakest) topology such that the projections to the factors are continuous.
A topological space is called connected if it does not split into a disjoint union of two open (hence closed) subsets. E.g. the real line and all intervals are connected, but the union of 2 disjoint open intervals is always disconnected. All connected subsets of the real line are open intervals (that may be empty and may be infinite) with, possibly, some of the ends attached. Any topological space splits into connected components that are maximal connected subsets. This gives a description of all open subsets of the real line: these are countable (or finite) disjoint unions of open intervals (or rays). The image of a connected space under a continuous map is also connected. This is a generalization of the Intermediate Value Theorem from Calculus. Connected components are always closed, but not always open. A counterexample is the set of all rational numbers with the topology induced from the reals (which is the same as the order topology) --- all rationals are separate connected components, but they are not open. A space all of whose connected components are open, is called locally connected. To be locally connected, it is enough to have at least one connected neighborhood for each point.
A path connecting two points of a topological space is a continuous map from a segment to this space such that the ends of the segment get mapped to given two points. If every pair of points of a topological space can be connected by a path, then the space is said to be path connected. Any path connected space is connected. Any space splits to path components, but they are not necessarily closed. There are examples of connected spaces that are not path connected. E.g. the graph of sin(1/x) together with the y-axis. A space all of whose path components are open (in other words, any point has a path connected neighborhood) is called locally path connected. A connected locally path connected space is path connected.
A topological space is said to be compact if any open covering of it contains a finite subcovering. A subset of a finite dimensional Euclidean space is compact if and only if it is closed and bounded. Closed subsets of compact spaces are compact in the subspace topology, the product of two compact spaces is also compact. For a metric space X the compactness is equivalent to any of the following statements:
A continuous function on a compact metric space is uniformly continuos.
The image of a compact space under a continuous map is compact. It follows that a continuous function on a compact space always attains its maximum and minimum.
The space C(X,Y) of all continuous functions of a topological space X to a metric space Y carries the natural uniform metric. Convergence with respect to this metric is exactly the uniform convergence. If Y is complete, then the space C(X,Y) is also complete. It follows that if a sequence of continuous functions converges uniformly to some function, then the limit is continuous.
A compact subset of a metric space is always closed and bounded. But in infinite dimensional spaces, being closed and bounded is not enough for being compact. E.g. the unit ball in C([0,1],R) is not compact. The Arzela-Ascoli theorem gives necessary and sufficient conditions on a subset A of C(X,Rn) for being compact. Namely, A is compact if and only if
The Baire theorem states that in a complete metric space, the intersection of countably many open dense sets is dense ("dense" means that the closure is the whole space). We say that "almost all points" of a complete metric space (in the sense of the Baire category) satisfy a certain property, if the set of points with this property is the intersection of countably many open dense sets. For example,
Some set theoretic preparation for the proof of the Tychonoff theorem is here [pdf, ps]. This is about the axiom of choice, the Zorn lemma and the Zermelo theorem.
A filter on a set X is a collection of subsets of X such that
The existence of a nonprincipal unltrafilter allows to define the field of hyperreal numbers that are used in the non-standard analysis. Let us fix a nonprincipal ultrafilter on the set of natural numbers. Define a hyperreal number to be a sequence of real numbers. Two such sequences are said to be equal if they are equal element-wise on some "big" set (a set from our ultrafilter). In a sense, all properties of real numbers have exact analogs for hyperreals. We can find the difference only while coparing the hyperreals with the reals (that are represented as constant sequences). And it turns out that there are infinitely small hyperreals (infinitesimals) that are positive and smaller than any positive real number. Infinitesimals allow to define derivatives, integrals, etc. without using the limits.
Let us say that a collection of sets has the finite intersection property if the intersection of any its finite subcollection is nonempty. The compactness of a to pological space is equivalent to any the following statements:
Consider the Cartesian product X of infinitely many topological spaces Xa. The Tychonoff topology on X is the smallest topology such that the projections to all components Xa are continuous. This definition is the same as that for finite products. But in the case of infinite product, this topology is really small (coarse). An open set in X is an open set in some finite subproduct cross all the other components. Thus any open set is very big: it differs from the whole space by only finitely many components. Although the Tychonoff topology is very coarse, it turns out to be just the right topology for many purposes. The Tychonoff theorem states that the product of any number of compact spaces is compact in the Tychonoff topology. The product of any number of connected spaces is also connected.
Two subsets of a topological space can be separated by open neighborhoods if they have disjoint open neighborhoods. Separation axioms for topological spaces state that certain pairs of subsets can be separated by open neighborhoods. A space is called
The Urysohn lemma states that for a normal topological space X and two disjoint closed subsets A and B of it, there exists a continuous function from X to [0,1] that is 0 on A and 1 on B. This condition guarantees that normal spaces have enough continuous functions. Suppose that a topological space X is normal and has a countable basis. Then there exist a countable family of continuous functions on X taking values in the unit segment [0,1] and satisfying the following property: for any point of X and any open neighborhood of this point, there is a function from our family such that this function is positive at the given point and is identically zero outside of the given neighborhood. In other words, a countably family of continuous functions is enough to separate any point from any closed set not containing this point. This follows from the Urysohn lemma and from the existence of a countable basis. A countable family of continuous functions with the property discussed above, gives a metric on X and an embedding of X into l2. This fact is known as the Urysohn metrization theorem.
To prove that some topological space is metrizable, we need to make sure that the topology on it is not too large. The countability axioms serve that purpose. The first countability axiom states that every point has a countable basis of neighborhoods, i.e. a collection of neighborhoods such that for any neighborhood our point there is a subneighborhood from our collection. The second countability axiom states that there is a countable basis. Every metric space satisfies the first countability axiom, but there are metric spaces that do not admit a countable basis (e.g. real numbers with the discrete metric). The first countability axiom allows to use limits of sequences. For general topological spaces, the limit of every convergent sequence from some set belongs to the closure of this set. But the converse is not true. There can be some points in the closure that can not be reached by any countable sequence (so that one needs to take uncountable sequences, say, indexed by a large well-ordered set, to reach those points). An example is given by any well-ordered set (with respect to the order topology) containing uncountable initial segments. Any regular second countable set is normal. Hence in the Urysohn theorem, the normality assumption can be replaced with just regularity assumption.
A collection of sets is called locally finite if any point has a neighborhood intersecting only finitely many sets from this collection. A countably locally finite collection is a countable union of locally finite collections. The Nagata-Smirnov metrization theorem (which we did not prove in class) gives necessary and sufficient conditions on a topological space to be metrizable. These conditions are formulated in purely topological terms. Namely, a topological space is metrizable if and only if it is regular and has a countably locally finite basis.
One collection of sets is said to be a refinement of another collection, if every set from the first collection is contained in some set from the second collection. A Hausdorff topological space is paracompact, if any open covering of it admits a locally finite refinement that covers the space. Clearly, any Hausdorff compact space is paracompact. By a very nontrivial theorem due to Stone, every metric space is also paracompact. Paracompact spaces are normal, in particular, regular. Any open covering of any paracompact space admits a partition of unity, i.e. a family of continuous functions that take values in [0,1], sum up to 1 and have small enough supports (the support of any function from this family must lie in some open set from the given open covering). Partitions of unity are very useful in geometry and topology, e.g. they help to prove embedding theorems. As an example of an embedding theorem, we proved that every compact topological manifold can be embedded into some affine space (a topological space with a countable basis is said to be a topological n-manifold if every point of this space has a neighborhood homeomorphic to Rn).
We started with examples of good topological spaces. These include: Euclidean spaces, spheres, real projective spaces, complex projective spaces, etc. Next we defined homotopy of paths (or a continuous deformation). Two paths are homotopic if there exists a homotopy between them. The relation of being homotopic is an equivalence relation. The equivalence classes with respect to this relation are called homotopy classes. A loop in a topological space is a path that begins and ends at the same point (this point is called the base point). Let us fix a base point. The homotopy classes of loops with this base point form so called the fundamental group of the topological space. The group structure is given by "multiplication of two loops" (the product of two loops is the loop formed by first going along one loop, and then going along the other loop). A continuous map from one path connected space to another path connected space is called a covering if it looks locally, over a small neighborhood, like the standard projection of the direct product of this neighborhood with a discrete space onto this neighborhood. Examples of coverings: