Category:Definitions/Examples of Topologies

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This category contains definitions of examples of topologies.

Let $S$ be a set.

A topology on $S$ is a subset $\tau \subseteq \powerset S$ of the power set of $S$ that satisfies 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 $\tau$.             

If $\tau$ is a topology on $S$, then $\struct {S, \tau}$ is called a topological space.

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


This category has the following 35 subcategories, out of 35 total.




Pages in category "Definitions/Examples of Topologies"

The following 111 pages are in this category, out of 111 total.