📂Topology

What is a Topological Space?

Definition

Topological Space ¹

Given a set $X$, if $\mathscr{T} \subset \mathscr{P} (X)$ satisfies the following three conditions for $T \in \mathscr{T}$, then $\mathscr{T}$ is called a topology of $X$, and $\left( X , \mathscr{T} \right)$ is called a topological space.

• (i): $$\emptyset , X \in \mathscr{T}$$
• (ii): $$\displaystyle \bigcup_{ \alpha \in \forall } T_{\alpha} \in \mathscr{T}$$
• (iii): $$\displaystyle \bigcap_{ i= 1}^{n} T_{i} \in \mathscr{T}$$

Restating conditions (i)~(iii) in words gives the following:

• (i): $\mathscr{T}$ contains the empty set $\emptyset$ and the whole set $X$.
• (ii): The union of elements of $\mathscr{T}$ belongs to $\mathscr{T}$.
• (iii): The finite intersection of elements of $\mathscr{T}$ belongs to $\mathscr{T}$.

Open Sets and Closed Sets ²

1. $O \in \mathscr{T}$ is defined as an open set.
2. For $C \subset X$, if $ X \setminus C \in \mathscr{T}$, then $C$ is defined as a closed set.
3. A set that is both open and closed is called a clopen set.

Explanation

Topological Space

By definition, $\mathscr{T}$ is closed under $\cup$ and $\cap$, so one might think of algebra, but such a definition alone makes it hard to find algebraic properties.

Just as a vector in a vector space is not 'a quantity with force and direction' as learned in high school but simply becomes a vector once it satisfies the conditions, the topology of a topological space is generalized as merely a collection of subsets satisfying the conditions.

Open Sets and Closed Sets

Along with defining the topology, openness and closedness are also newly defined. In the existing metric space, they were defined intuitively as concepts extending from open intervals and closed intervals, but a general topology uses sets, so it can create abstract and fantastic spaces.

Looking at the definition, openness is defined completely anew by borrowing the concept of topology, while closedness can be seen to be almost the same as in a metric space.

According to the definitions of topology, openness, and closedness, the following properties can be easily verified.

Theorem

• [1-1]: $\displaystyle \bigcup_{ \alpha \in \forall } O_{\alpha} \in \mathscr{T}$ is an open set.
• [1-2]: $\displaystyle \bigcap_{ i= 1}^{n} O_{i} \in \mathscr{T}$ is an open set.
• [2-1]: $\displaystyle \bigcap_{ \alpha \in \forall } C_{\alpha} \in \mathscr{T}$ is a closed set.
• [2-2]: $\displaystyle \bigcup_{ i= 1}^{n} C_{i} \in \mathscr{T}$ is a closed set.
• [3]: $\emptyset$ and $X$ are both open and closed sets.

Example

Let us look at the following example to get a feel for topology.

For $X:=\left\{ a,b,c,d \right\}$, show that $\mathscr{T} : = \left\{ \emptyset , \left\{ b \right\} , \left\{ a, b \right\} , \left\{ b,c \right\} , \left\{ a,b,c \right\} , \left\{ a,b,c,d \right\} \right\}$ is a topology of $X$.

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• (i): $\emptyset \in \mathscr{T}$ and $\left\{ a,b,c,d \right\} =X \in \mathscr{T}$.
• (ii): Except for the whole set $X$, $d$ is not used, and $\left\{ a,b,c \right\} \in \mathscr{T}$.
• (iii): Except for the empty set $\emptyset$, all share $b$, and $\left\{ b \right\} \in \mathscr{T}$.

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