# Topological space

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## Definition

A topological space is a set $X$ coupled with a "topology", [ilmath]\mathcal{J} [/ilmath] on $X$. We denote this by the ordered pair [ilmath](X,\mathcal{J})[/ilmath].

• A topology, [ilmath]\mathcal{J} [/ilmath] is a collection of subsets of [ilmath]X[/ilmath], $\mathcal{J}\subseteq\mathcal{P}(X)$ with the following properties[1][2][3]:
1. Both $\emptyset,X\in\mathcal{J}$
2. For the collection $\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J}$ where $I$ is any indexing set, $\cup_{\alpha\in I}U_\alpha\in\mathcal{J}$ - that is it is closed under union (infinite, finite, whatever - "closed under arbitrary union")
3. For the collection $\{U_i\}^n_{i=1}\subseteq\mathcal{J}$ (any finite collection of members of the topology) that $\cap^n_{i=1}U_i\in\mathcal{J}$
• We call the elements of [ilmath]\mathcal{J} [/ilmath] "open sets", that is [ilmath]\forall S\in\mathcal{J}[S\text{ is an open set}] [/ilmath], each [ilmath]S[/ilmath] is exactly what we call an 'open set'

As mentioned above we write the topological space as $(X,\mathcal{J})$; or just $X$ if the topology on $X$ is obvious from the context.

## Comparing topologies

Given two topological spaces, [ilmath](X_1,\mathcal{J}_1)[/ilmath] and [ilmath](X_2,\mathcal{J}_2)[/ilmath] we may be able to compare them; we say:

Terminology If Comment
[ilmath]\mathcal{J}_1[/ilmath] coarser[2]/smaller/weaker [ilmath]\mathcal{J}_2[/ilmath] [ilmath]\mathcal{J}_1\subseteq\mathcal{J}_2[/ilmath] Using the implies-subset relation we see that [ilmath]\mathcal{J}_1\subseteq\mathcal{J}_2\iff\forall S\in\mathcal{J}_1[S\in\mathcal{J}_2][/ilmath]
[ilmath]\mathcal{J}_1[/ilmath] finer[2]/larger/stronger [ilmath]\mathcal{J}_2[/ilmath] [ilmath]\mathcal{J}_2\subseteq\mathcal{J}_1[/ilmath] Again, same idea, [ilmath]\mathcal{J}_2\subseteq\mathcal{J}_2\iff\forall S\in\mathcal{J}_2[S\in\mathcal{J}_1][/ilmath]