📂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 the 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): $$\displaystyle \bigcap_{ i= 1}^{n} T_{i} \in \mathscr{T}$$

Explained in words, conditions (i)~(iii) are as follows:

• (i): $\mathscr{T}$ includes the empty set $\emptyset$ and the entire 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 and Closed Sets ²

1. $O \in \mathscr{T}$ is defined as an Open Set.
2. For $C \subset X$, if $X \setminus C \in \mathscr{T}$ is true, then $C$ is defined as a Closed Set.
3. If a set is both an open set and a closed set, it is called a Clopen Set.

Explanation

Topological Spaces

The definition suggests that $\mathscr{T}$ is closed under $\cup$ and $\cap$, evoking thoughts of algebra, but this definition alone makes it hard to find algebraic properties.

Just as a vector in a vector space becomes a vector not just by having force and direction as learned in high school but by satisfying certain conditions, the topology of topological space is generalized as the set of subsets that satisfy certain conditions.

Open and Closed Sets

With the definition of topology, the concepts of openness and closure are redefined. In conventional metric spaces, these concepts were defined intuitively, following from intervals open and closed. However, general topology uses sets, which can create abstract and bizarre spaces.

The definition reveals that while openness is completely redefined via the concept of topology, closure remains almost the same as in metric spaces.

Based on the definitions of topology, openness, and closure, the following properties can be easily verified.

Theorems

• [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.

Examples

Let’s get a feel for topology with the following example.

Show that for $X:=\left\{ a,b,c,d \right\}$, $\mathscr{T} : = \left\{ \emptyset , \left\{ b \right\} , \left\{ a, b \right\} , \left\{ b,c \right\} , \left\{ a,b,c \right\} , \left\{ a,b,c,d \right\} \right\}$ is the 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 entire 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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