Difference between revisions of "Topological space/Definition"

Latest revision as of 18:09, 20 April 2016

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]^[2]^[3]:

Both $\emptyset,X\in\mathcal{J}$
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")
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.

@@ Line 5: / Line 5: @@
 # Both <math>\emptyset,X\in\mathcal{J}</math>
-# For the collection <math>\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J}</math> where <math>I</math> is any [[indexing set]], <math>\cup_{\alpha\in I}U_\alpha\in\mathcal{J}</math> - that is it is closed under union (infinite, finite, whatever - "closed under ''arbitrary'' union")
+# For the [[collection]] <math>\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J}</math> where <math>I</math> is any [[indexing set]], <math>\cup_{\alpha\in I}U_\alpha\in\mathcal{J}</math> - that is it is closed under union (infinite, finite, whatever - "closed under ''arbitrary'' union")
 # For the collection <math>\{U_i\}^n_{i=1}\subseteq\mathcal{J}</math> (any ''finite'' collection of members of the topology) that <math>\cap^n_{i=1}U_i\in\mathcal{J}</math>