List of topological properties

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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
Closure	Let [ilmath]A\in\mathcal{P}(X)[/ilmath] be given. The closure of [ilmath]A[/ilmath], denoted [ilmath]\overline{A} [/ilmath] is defined as follows: [ilmath]\overline{A}:\eq\bigcap\left\{C\in\mathcal{C}(X)\ \big\vert\ A\subseteq C\right\} [/ilmath] - where [ilmath]\mathcal{C}(X)[/ilmath] denotes the set of closed sets of [ilmath]X[/ilmath] Informally, it is the smallest closed set containing [ilmath]A[/ilmath]. Note that the largest closed set c	Probably something with limit points	See also: Interior Boundary
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 we may write this as: [ilmath]\forall p\in X-A\left[\neg\left(\exists U\in\mathcal{J}[p\in U\wedge U\subseteq A\right)\right][/ilmath] [ilmath]\iff\forall p\in X-A\forall U\in\mathcal{J}[\neg(p\in U\wedge U\subseteq A)][/ilmath] [ilmath]\iff\forall p\in X-A\forall U\in\mathcal{J}[(\neg(p\in U))\vee(\neg(U\subseteq A))][/ilmath] - by the negation of logical and [ilmath]\iff\forall p\in X-A\forall U\in\mathcal{J}[p\notin U\vee U\nsubseteq A][/ilmath] - of course by the implies-subset relation we see [ilmath](A\subseteq B)\iff(\forall a\in A[a\in B])[/ilmath], thus: [ilmath]\iff\forall p\in X-A\forall U\in\mathcal{J}\big[p\notin U\vee(\exists q\in U[q\notin A])\big][/ilmath] TODO: Tidy this up
Interior	[math]\text{Int}(A,X):\eq\bigcup_{U\in\{V\in\mathcal{J}\ \vert\ V\subseteq A\} }U[/math]		Could be union of all interior points, see here
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