General Topology - Intro and Continuous Functions

I hate topology. --- By Woziji Shuode

Topology

Definition 1. Topology

A topology on a set X is a collection of its subsets , which has the following three properties:

1. and ;
2. For arbitrary subcollection of collection , the union of is in too;
3. For any finite subcollection of , the intersection of the elements of is in too.

When we are talking about some topological properties of any set , we are using a certain topology . In that case, we say " is a topological space with topology ". In the following content, we default topological space is with topology .

Definition 2. [Open]

A subset of is said to be open if it is in the topology .

Till now, the definition seems to be too abstract to understand, therefore let's give an example.

Example 3. Suppose we have a metric space , then has a topology , which is the set of all its open subsets.(Here, we use the definition of open from Basic Topology) Then:

1. and are both open;
2. For arbitrary collection of open sets, their union is open;
3. For finite collection of open sets, their intersection is open.

Some topology is special:
Definition 4. [Discrete topology, Indiscrete / Trivial topology, Finite complement topology]

1. The collection of all subsets of is called discrete topology.
2. is called indiscrete topology or trivial topology.
3. let . is a topology too.

Topologies are naturally collections of subsets, therefore there are clearly relationships of inclusion between them. For topologies, we have other words to describe them.

Definition 5. [Comparable, (Strictly) Finer, (Strictly) Coaser] For two topologies and :

1. If either or , we called and comparable;
2. If , we call is finer than .
3. If , we call is coarser than ;
4. If and are comparable while , they are either strictly finer or strictly coarser.

Basis

Despite of topology, we also have a concept basis, which is a bit "looser" than topology.

Definition 6. Basis

A basis of a set is a collection of subsets of such that:

1. , such that ;
2. , such that and .

As its name, basis are the bases of topologies. If is the basis of topology , we say is generated from .

Lemma 7. 1. is called "generated by ". In other words, it's the collection of all unions of elements of .
From this Lemma, we have the topology generated by a certain basis is unique. 2. Let be a collection of open sets of w.r.t. topology , such that for any and for any point in (), such that . Then we say is a basis for the topology of .
Different from , a topology can have multiple bases. For example, is a basis for itself.

Definition 8. [Subbasis]

A subbasis of a topology is a collection of subsets of , such that the union of all elements of equals to . The topology generated by is defined as the collection of all unions of finite intersections of elements of .

The subbasis for can be seen as "the smallest set of subsets of , such that after I completed its topology properties, it becomes ".

Subspace Topology

Definition 9. [Subspace Topology, Subspace Basis]

Let be a space with topology , with basis . Let be a subset of , then

1. is a topology of ;
2. is a topology of ;

Closed sets, Limit points

Definition 10. [Closed set] A set is said to be closed if (or ) is open.

Lemma 11. 1. and are closed; 2. Arbitrary intersections of closed sets are closed; 3. Finite union of closed sets are closed;

Lemma 12. 1. Let be a subspace of a topological space , then is closed in if and only if there is a set closed in such that ; (Important for problems!) 2. Let be a closed subspace of . A subset is closed in if and only if it is closed in .

Definition 14. [Interior, Closure]

Suppose is a topological space and , then we have:

1. The interior of is . That is, the interior of is the union of all open sets contained in ;
2. The closure of is . That is, the closure of is the intersection of all closed sets containing ;

Theorem 15. Let be a subset of a topological space with basis , then:

1. iff every open set containing intersects ;
2. iff every containing intersects

Definition 16. [Neighborhood, Limit point] Let be a subset of topological space , then

1. For , every open set containing is called a neighborhood of ;
2. Let be a subset of a topological space with basis , then is a limit point of if every neighborhood of satisfies . The set of limit points of is written as /

Theorem 17. .

Theorem 18. Let be a subset of topological space , then T.T.F.E. :

1. is closed;
2. ;
3. .

Sequential Limits and Hausdorff Space

Definition 19. [Convergence]

Let be a sequence of points of , then we say converges to if for any neighborhood of , there exists such that for any , .

Definition 20. [Hausdorff Space]

A topological subspace is called Hausdorff space if for any , has a neighborhood , has a neighborhood , such that .

Theorem 21. If is a Hausdorff space, then T.T.F.E. :

1. Every finite subset is closed;
2. If converges, then its limit is unique;
3. For any subset , is a limit point of iff every neighborhood contains infinitely many points of .

Continuous Functions

Definition 22. [Continuous]

Let , be two topological spaces. Then a function is continuous iff for any open subset , is open in .

Definition 23. [Homeomorphism]

Let , be two topological spaces. Then a bijection is homeomorphism if both and is continuous.