Def/Topological space

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Definition
Basis

Definition of Topological space: A topological space is an ordered pair $(X,T)$, where $X$ is a set, and $T$ is set of subsets of $X$, which satisfies the following axioms:

If $S \subset T$, then $\bigcup S \in T$. This includes the "empty union", i.e., if $S = \emptyset$, then $\bigcup S = \emptyset \in T$.
If $S \subset T$, and $S$ is finite, then $\bigcap S \in T$. This includes the "empty intersection", i.e., if $S = \emptyset$, then we define $\bigcap S = X$, and so $X \in T$.

$T$ is called a topology on $X$, and the elements of $T$ are called the open subsets of $X$ (for the topology $T$).

Logical Connections

This definition logically relies on the following definitions and statements: Def/Subset, Def/Finite (set)

The following statements and definitions logically rely on the material of this page: Def/Compact, and Def/Continuous function (in topology)

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Topology

Def/Topological space

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	Suppose that $b_1, b_2 \in B$, and $x \in b_1 \cap b_2$.
	Then $b_1, b_2 \in T$, so $b_1 \cap b_2 \in T$. Hence, there exists a subset $S \subset B$, such that $b_1 \cap b_2 = \bigcup S$. Hence, there exists an element $b_3 \in S$, such that $x \in b_3$. Since $b_3 \in S$, and $b_1 \cap b_2 = \bigcup S$, we find that $b_3 \subset b_1 \cap b_2$.
	It follows that $x \in b_3 \subset b_1 \cap b_2$.

	Suppose that $x \in X$.
	Since $X \in T$, we find that there exists a subset $S \subset B$, such that $X = \bigcup S$. Hence, there exists an element $b \in S$, such that $x \in b$.
	Therefore, $x \in b$ and $b \in B$.