S. D. V. Stephens

notes / Studies in Real and Complex Analysis / chapter 1: Metric and Topological Spaces

3 Limit Points and Hausdorff Spaces

· Denis Auroux

Original Professor Auroux’s handwritten notes (PDF), behind a password for now:

Remark (Last time).

  • A topology on a set is a collection of open sets such that (1) , are open; (2) , open open; (3) open open.
  • A basis for is such that open sets = unions of elements of (or: is open such that ).

This can simplify the description of a topology, and also verifications of continuity, convergence, etc. E.g., if have topologies with bases : (For the second equivalence, going back up: write ; then .) Compare the definition of continuity via balls in metric spaces.

Subspaces and products

  • Subspace topology on : .

Definition 3.1. Let , be topological spaces. The product topology on is the topology generated by the basis

(Check that is a basis. Why isn’t already a topology?)

In the plane X times Y, a blue rectangle U_1 x V_1 and a red rectangle U_2 x V_2 overlap; their intersection, shaded, is again a rectangle, built from the overlaps of the intervals on each axis.
  • When are metric spaces, this is also a metric topology, defined e.g. by (check! Key observation: ). Or in fact define the same topology on (see HW for the case of ). So this gives the usual topology on .
  • is continuous and are continuous.
  • A sequence in converges, , and .
  • Similarly for finite products . For infinite products there are several different natural topologies; see next week.

Homeomorphisms

When are two topological spaces “the same”?

Definition 3.2. are homeomorphic if there exist continuous maps and such that , .

Equivalently, a homeomorphism is a continuous bijection such that is continuous. I.e., a bijection under which .

Remark.

  • A continuous bijection need not be a homeomorphism. E.g. take with two topologies, strictly finer than (). Then is a bijection, continuous since , but not a homeomorphism.
  • Say a metric space is bounded if . This is not a topological property: e.g. , is a homeomorphism ( and are continuous), so is homeomorphic to (or any open interval in ).
Graph of tan x on the open interval from minus pi over 2 to pi over 2, with dashed vertical asymptotes at both ends, showing a homeomorphism from this interval onto the real line.

Closed sets

Definition 3.3. A subset of a topological space is closed if is open.

Warning: subsets can be both closed and open, e.g. and , or neither (e.g. or in ).

The axioms of open sets imply:

  • , are closed;
  • arbitrary intersections of closed sets are closed;
  • finite unions of closed sets are closed.

Definition 3.4. Let be any subset. We define:

  1. the closure of : = the smallest closed set containing (, and is closed since it’s an intersection of closed sets);
  2. the interior of : = the largest open set contained in (open);
  3. the boundary of : .
Four versions of a disc-shaped set with a hair attached: A (boundary partly included, hair with open end), its closure (solid boundary, hair with closed end), its interior (dashed boundary, no hair), and its boundary (circle plus hair, unfilled).

Example 3.5. , usual topology , , .

Remark.

  • is closed iff ; open iff .
  • and .

Definition 3.6. Say is a neighborhood of if is open and .

Proposition 3.7.

  1. iff contains a neighborhood of .
  2. iff every neighborhood of intersects nontrivially.

Proof. Check this! (1) follows from the definitions: open such that .

(2) follows from (1) and : open, . ∎

Definition 3.8. Say is dense if (i.e. every nonempty open subset of intersects nontrivially).

Example 3.9. is dense in (for the usual topology).

Closed sets and limit points

Definition 3.10. is a limit point of if for every neighborhood , .

A shaded set A with three points and dashed neighbourhoods: a point inside A and a point on its edge are limit points, while a point away from A whose neighbourhood misses A is not.

Example 3.11. In , is a limit point of and of . is not a limit point of , but is.

Proposition 3.12. .

Proof. by definition, so it’s enough to consider points not in . If , then for every neighborhood , is a limit point if the left-hand side is always , and if the right-hand side is always . So iff is a limit point. ∎

Corollary 3.13. is closed iff contains all of its limit points.

Warning: what is the connection between limit points and limits of sequences?

Recall: a sequence in converges to if neighborhood of , such that .

Proposition 3.14. Let and .

  • If sequence in such that , then .
  • If sequence in with for infinitely many and , then is a limit point of .

Proof. Any neighborhood contains for all large , hence contains points of (distinct from in the second case). ∎

The converse is true in metric spaces: if (resp. is a limit point of ) then (resp. with ), so there is a sequence in such that .

This holds more generally in spaces whose points have countable bases of neighborhoods (i.e. neighborhoods such that for every neighborhood , such that ), but not in arbitrary topological spaces!

Hausdorff spaces

In a metric space, a sequence converges to at most one limit. This is not true in an arbitrary topological space!

Example 3.15. with the finite complement topology: open subsets are and . Let be a sequence in with all distinct. Then , every neighborhood contains all but finitely many of the , hence such that . Thus the sequence converges to every point of !

To avoid such pathological behavior:

Definition 3.16. A topological space is Hausdorff (or ) if , neighborhoods , such that .

Example 3.17.

  1. Any metric space is Hausdorff: given , choose ; then are disjoint neighborhoods of .

    Two points x_1 and x_2 with disjoint open balls B_epsilon(x_1) and B_epsilon(x_2) around them.

  2. The finite complement topology on is not Hausdorff, since any two nonempty open sets intersect (in infinitely many points).

  3. The discrete topology is always Hausdorff ( are disjoint neighborhoods of ).

  4. One can show: Hausdorff, the subspace topology is Hausdorff; Hausdorff Hausdorff. (Homework!)

Notes not reached in this lecture, covered at the start of Lecture 4

Theorem 3.18. If is Hausdorff then every sequence in converges to at most one limit.

Proof. Assume converges to , and let . Choose disjoint neighborhoods , . Since , such that , . Hence for , so the sequence doesn’t converge to . ∎

Remark.

  • There’s in fact a whole hierarchy of “separation axioms”. E.g. a weaker one is: a topological space is if , neighborhood such that . Equivalently: is is closed in (exercise!). Hausdorff () , but e.g. (, finite complement topology) is but not Hausdorff.
  • Hausdorff spaces are fairly nice to work with, and we will generally be working with this assumption. There are more subtle reasons why not every Hausdorff topology comes from a metric, but one can give pretty good criteria for a topology to be metrizable involving further separation conditions (“normal” or ) (plus a countability condition). We’ll see the Urysohn metrization theorem.

Manifolds and CW complexes

Metric spaces are nice, but they can still be pretty nasty. (We’ll see conditions such as local connectedness, local compactness etc. come up.) Algebraic topologists like to focus on even nicer spaces. For example:

Definition 3.19. An -dimensional topological manifold is a topological space such that every point has a neighborhood homeomorphic to (or equivalently, to an open ball in ).

Example 3.20. is a 1-dimensional topological manifold; the sphere and the torus in are 2-dimensional topological manifolds.

A circle S^1 in the plane, and a sphere and a torus in three-dimensional space, as examples of topological manifolds.

Example 3.21.

A Y-shaped graph made of three segments meeting at a single vertex, which is not a manifold because of the vertex.

This Y-shaped graph isn’t a topological manifold (the vertex looks wrong), but it is part of a more general class of spaces called CW complexes, built by attaching “cells” (closed balls of dimension ) onto each other inductively.

We’ll see more on this later when we get to algebraic topology. In decreasing order of generality: