[[Topology MOC]]
# Topological space
Abstractly, a **topological space** $(X, \mathcal T)$ consists of a set $X$ 
and a collection of subsets $\mathcal T \sube \mathcal{P}(X)$
such that[^br] #m/def/topology 
1. $\mathcal T$ contains at least $\0$ and $X$.
2. Any finite or infinite union of subsets in $\mathcal T$ is also in $\mathcal T$.
3. Any finite intersection of subsets in $\mathcal T$ is also in $\mathcal T$.

where $\mathcal T$ is called a **topology** on $X$,
and is said to contain **open subsets** of $X$.
A subset of $X$ is called **closed** iff its compliment is open.
Thus, in any topological space $(X, \mathcal{T})$ the subsets $X$ and $\0$ are **clopen**^[Simultaneously open and closed.].

On any set $X$ we can easily form the [[Discrete topology]] $\mathcal{P}(X)$ (every set is **clopen**)
and the [[Trivial topology]] $\{ \0, X \}$.

Two topologies on the same space $X$ can be compared in terms of [[Coarseness and fineness of topologies]].

A topology can be generated by a [[Topological basis]].

[^br]: 2020, [[@bradleyTopologyCategoricalApproach2020|Topology: A categorical approach]], §0.1, p. 1

## Properties

- [[The intersection of topologies on a fixed set is again a topology]]
- [[Coarseness and fineness of topologies]]

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