Topological Preliminaries

Main References

Munkres. (2000). Topology (2nd Edition)

Topologies and Bases

Topology

Definition (topological set).

Let be a given set and a topology is a set (or, family) of subsets of . Then a tuple is a topological space if the followings hold:

and
is closed under unions:
is closed under countable intersections:

Here, is an open set of .

Proposition (intersection of topology).

The intersection of topologies is also a topology.

Proof.Let be an arbitrary collection of topologies on . and define . We show that satisfies the three conditions of the topology.

trivially, , since for all , we have .
Let . Then for all , . And by the properties of topology, . Then, by implication, we have .
Let the arbitrary . Then since , for every , we have . Then, since is topology, we have for all . Thus we have .

Thus is a topology. □

Remark that the concept of an open set is different from an open interval. Denote the set of all open sets in the real space and its unions as then by ^37c21c Definition 1 (topological set), it can be shown that constructs a topological space over . However, for the set of all intervals in including closed intervals and others (such as or ), as is a topological space, then every elements in is an open set, i.e. closed intervals are also open sets.

Intuitively, topological space makes it possible to define an open set even in a non-metric space. Below, we can see that ^37c21c Definition 1 (topological set) imitates the ^fa78c6 Remark 3 (properties of open set in metric space).

Remark (properties of open set in metric space).

An open set of a metric space satisfies the following conditions:

and are open sets
If are open and , then is open.
If are open and , then is open.

Bases and Local Bases

Definition (base).

Let be an arbitrary set. A collection of subsets of is base if the following two conditions hold:

covers , i.e. .
and , there exists a such that .

In topology, a base for the topology is a family of open sets such that every open set of the topology can be generated as the union of some subfamily of . These elements of are called basic open sets. Clearly, is contained in the topology .

Below, we look into some theorem on the relationships between the topology and its base. Note that to each base, there exists only a single topology that it generates,

Lemma (topology generated by base).

Let be a base of . Now consider the collection of subsets of satisfying the following properties:

contains s.t.

Then, is a topology on , and it is called the topology generated by the base .

Proof.Remark the three conditions from ^37c21c Definition 1 (topological set). We prove that satisfies these three conditions.

By definition, . Also, contains such that and because by ^5ac2e4 Definition 4 (base), we have
For any collection , denote .
1. if , then trivially .
2. if , let any element be . Then for some , we have . Also, as , as , we therefore have .
For any , let .
1. if , then trivially .
2. if , then any elements is and .
  1. since , where and thus .
  2. similarly, we have where .
  3. also, we have
  4. Since we have , meaning that .

Therefore, is a topology on . □

The converse also holds.

Lemma (condition for a generating base).

Let be a topology on , and be a collection of subsets of . Then, is a base on that generates if the followings are satisfied:

is contained in : any subset in is also in .
For any nonempty and , there exists a

Examples of Topologies

Order Topology

Metric Topology

Product Topology

Euclidean Topology

Closures and Interiors

Definition (closed set).

Let be a topological space. A set is closed if for some open set .

Proposition (properties of closed set).

Let be a topological space. Since is said to be closed if , we have the following properties:

and are closed.
For any arbitrary collection of closed subsets of , is closed.
For any finite collection of closed subsets of , is closed.

Proof.(1) From ^37c21c Definition 1 (topological set), we have . Thus we have and , implying that and are closed.

(2) Since are closed for any , we have . Thus by the closedness under arbitrary unions of ^37c21c Definition 1 (topological set), we have . Therefore, is closed.

(3) Since are closed, we have for . Thus by the closedness under finite intersections of ^37c21c Definition 1 (topological set), we have . Therefore, is closed. □

Now we employ a similar concepts from the metric spaces.

Definition (limit point and convergence).

Let be a topological space, and consider a point .

any open set such that is called a neighborhood of .
is a limit point of if for any neighborhood , we have .
the set of all limit points of is denoted as and called a derived set:
for any sequence , we say converges to if

Definition (closure and interior).

Let be a topological space and is contained in at least one closed set.

is an interior point of if there exists an open set such that .
is an interior of if
is a closure of if
is a boundary point of if
is a boundary of if
is dense in if .
is nowhere dense if .

Remark (closure and boundary).

Let be a topological space. Then we have note that and are disjoint by the definition.

Note that we can also define the closure and interior using open and closed sets.

Remark (another definition of closure and interior).

Let be a topological space, and . Note that the set is nonempty. Thus we can re-define the definitions as follows:

Topological Preliminaries

Topologies and Bases

Topology

Bases and Local Bases

Examples of Topologies

Order Topology

Metric Topology

Product Topology

Euclidean Topology

Closures and Interiors

Countability and Separability

First and Second Countability

Second Countability and Separability

Compact Sets and Spaces

Compact Sets and Related Concepts

Compact Sets in Hausdorff Spaces

Bolzano-Weierstrass Theorem

Continuous Functions

Characteristics of Continuity

Homeomorphisms

Semicontinuity

Connectedness