Topological space

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Definition

A topological space is a set

$$X$$ coupled with a "topology", $\mathcal{J}$ on $$X$$ . We denote this by the ordered pair $(X,\mathcal{J})$.

• A topology, $\mathcal{J}$ is a collection of subsets of $X$, $$\mathcal{J}\subseteq\mathcal{P}(X)$$ with the following properties:

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 $\mathcal{J}$ "open sets", that is $\forall S\in\mathcal{J}[S\text{ is an open set}]$, each $S$ 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, $(X_1,\mathcal{J}_1)$ and $(X_2,\mathcal{J}_2)$ we may be able to compare them; we say:

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

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This page requires references, it is on a to-do list for being expanded with them.

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Need references for larger/smaller/stronger/weaker, Check Introduction To Topology - Mendelson, Addendum: investigate relating this to a poset (easy enough - not very useful / lacking practical applications)

Examples

• Every metric space induces a topology, see the topology induced by a metric space
• Given any set $X$ we can always define the following two topologies on it:
  1. Discrete topology - the topology $\mathcal{J}=\mathcal{P}(X)$ - where $\mathcal{P}(X)$ denotes the power set of $X$
  2. Trivial topology - the topology $\mathcal{J}=\{\emptyset, X\}$

See Also

• Topology (subject)
• Topological property theorems
• Topology induced by a metric