Difference between revisions of "Topological space"

Revision as of 04:28, 8 April 2015

A topological space is a set

$X$ coupled with a topology on

$X$ denoted

$\mathcal{J}\subset\mathcal{P}(X)$ , which is a collection of subsets of

$X$ with the following properties:

Both $\emptyset,X\in\mathcal{J}$
For the collection $\{U_\alpha\}_{\alpha\in I}\subset\mathcal{J}$ 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)
For the collection $\{U_i\}^n_{i=1}\subset\mathcal{J}$ (any finite collection of members of the topology) that $\cap^n_{i=1}U_i\in\mathcal{J}$

We write the topological space as

$(X,\mathcal{J})$ or just

$X$ if the topology on

$X$ is obvious.

The elements of

$\mathcal{J}$ are defined to be "open" sets.

EVERY BOOK WITH TOPOLOGY IN THE NAME AND MANY WITHOUT

@@ Line 1: / Line 1: @@
+==Definition==
 A topological space is a set <math>X</math> coupled with a topology on <math>X</math> denoted <math>\mathcal{J}\subset\mathcal{P}(X)</math>, which is a collection of subsets of <math>X</math> with the following properties:
@@ Line 9: / Line 11: @@
 The elements of <math>\mathcal{J}</math> are defined to be "[[Open set|open]]" sets.
+==See Also==
+* [[Topology]]
+==References==
+ EVERY BOOK WITH TOPOLOGY IN THE NAME AND MANY WITHOUT
 {{Definition|Topology}}