List of topological properties

Stub grade: A*

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Needs linking in to places. Because density is SPRAWLED all over the place right now

Index

Here [ilmath](X,\mathcal{J})[/ilmath] is a topological space or [ilmath](X,d)[/ilmath] is a metric space in the definitions.

Property	Topological version	Metric spaces version	Comments
Dense set	For [ilmath]A\in\mathcal{P}(X)[/ilmath] we say [ilmath]A[/ilmath] is dense in [ilmath]X[/ilmath] if: [ilmath]\forall U\in\mathcal{J}[U\cap A\neq\emptyset][/ilmath]	For [ilmath]A\in\mathcal{P}(X)[/ilmath] we say [ilmath]A[/ilmath] is dense in [ilmath]X[/ilmath] if: [ilmath]\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap A\neq\emptyset][/ilmath] Caveat:This is given as equiv to density by - also obviously follows from it!	See also: nowhere dense set
	Equivalent statements
	The following are equivalent to the definition above. [ilmath]\text{Closure}(A)[/ilmath][ilmath]\eq X[/ilmath] [ilmath]X-A[/ilmath] contains no (non-empty) open subsets of [ilmath]X[/ilmath] Symbolically: [ilmath]\forall U\in\mathcal{J}[U\nsubseteq X-A][/ilmath], which we can easily manipulate to get: [ilmath]\forall U\in\mathcal{J}\exists p\in U[p\notin X-M][/ilmath] [ilmath]X-A[/ilmath] has no interior points (see below) Symbolically.... TODO: Can't be bothered right now, include later
Interior point	For a set [ilmath]A\in\mathcal{P}(X)[/ilmath] and [ilmath]a\in A[/ilmath], [ilmath]a[/ilmath] is an interior point of [ilmath]A[/ilmath] if: [ilmath]\exists U\in\mathcal{J}[a\in U\wedge U\subseteq A][/ilmath]	For a set [ilmath]A\in\mathcal{P}(X)[/ilmath] and [ilmath]a\in A[/ilmath], [ilmath]a[/ilmath] is an interior point of [ilmath]A[/ilmath] if: [ilmath]\exists\epsilon>0[B_\epsilon(a)\subseteq A][/ilmath] Caveat:Basically follows from topological definition, these are closely related