Topological Spaces

Topology

Topological Space

A topology on a set is a collection of subsets of , which we agree to call the “open sets”, such that

1. and are open
2. the intersection of finitely many open sets is open
3. arbitrary unions of open sets are open.

The pair is called a topological space.

e.g. The topology induced by a metric: in any metric space the collection of all open sets forms a topology.

Closed Sets

In a topological space , a set is called closed if is open. i.e. .

Discrete Topology & Trivial Topology

A topology is said to be discrete if all subsets are open. It is indiscrete or trivial if the only open sets are and .

Cofinite Topology & Cocountable Topology

A topology is cofinite if all open subsets are , , or the set whose complement is finite. A topology is cocountable if all open subsets are , , or the set whose complement is countable.

Zariski Topology on

A set is open if it is , , or its complement is the set of zeros of a polynomial with real coefficients.

Metrizable

Topologies need not come from metrics, but if there is, we say that is metrizable.

Lemma

Suppose that consists of more than one point. Then the indiscrete topology on is not metrizable.

Proof Assume the indiscrete topology on is induced by a metric on . Let with . Then . The set is an open subset of . Since , this set is not empty. And since this set is not all of . But and are the only open sets, yielding a contradiction.

Coarser and Finer

If and are two topologies on then we say that is coarser than if , that is contains fewer open sets than . In this situation, we also say that is finer than .

e.g. Given a set , the trivial topology is the coarsest/weakest topology on and the discrete topology is the finest/strongest topology on .

Bases and Sub-bases

Basis for a Topology

A basis for a topology on is a collection such that every set in is the union of some sets from .

e.g. Let be a metric space. Then is a basis for the metric topology on .

Theorem

Let be a topological space. Then is a basis for iff for any and any with there is such that .

Proposition

A collection of sets cannot be basis for two distinct topologies.

Proof Suppose that is a basis for both and . Then every set in is a union of sets in . Since , every set in is open in , this implies that . Similarly, we have .

Lemma If is a basis for then

• is the union of some sets from
• If then is the union of some sets from .

Thrm Let be a set and let be a collection of subsets of that satisfy property(B1) and property(B2). Then there is a unique topology on whose basis is . Its open sets are precisely the unions of sets from : Proof

Theorem

Let be a set. Let and be two topologies on with bases and respectively. The following are equivalent:

• .
• For each and each there is such that .

Proof

Sub-basis

A sub-basis for a topology on is a collection such that every set in is a union of finite intersections of sets from .

Subspaces and Finite Product Spaces

Subspace Topology

If is a topological space and , then the subspace topology on is We call a topological subspace of .

e.g. If we consider as a subspace of then the open sets in consist of all sets where is an open subset of . In particular, is open for every .

Lemma

Suppose that is a metric space with corresponding topology . If then subspace topology on corresponds to the topology on that arises from the metric space .

Proof

Proposition

Closed subset in closed subspace is closed in the original topological space.

Proof Let be a closed subspace of , and is closed in . Then for some open set in . It follows that . Thus, Since is closed in , is open. Therefore is open, and is closed in .

Product Topology

Suppose that and are two topological spaces. Then the product topology on is the topology with basis We call the topological product of and .

Thrm Let and be topological spaces with respective bases and . Then