Category:Definitions/Examples of Topological Spaces

From ProofWiki
Jump to navigation Jump to search

This category contains definitions of examples of topological spaces.

Let $S$ be a set.

Let $\tau$ be a topology on $S$.

That is, let $\tau \subseteq \powerset S$ satisfy the open set axioms:

\((\text O 1)\)   $:$   The union of an arbitrary subset of $\tau$ is an element of $\tau$.             
\((\text O 2)\)   $:$   The intersection of any two elements of $\tau$ is an element of $\tau$.             
\((\text O 3)\)   $:$   $S$ is an element of $\tau$.             

Then the ordered pair $\struct {S, \tau}$ is called a topological space.

The elements of $\tau$ are called open sets of $\struct {S, \tau}$.


This category has only the following subcategory.

Pages in category "Definitions/Examples of Topological Spaces"

The following 3 pages are in this category, out of 3 total.