List of topological properties

Stub grade: A*

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

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

Index

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

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