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 $(X,\mathcal{J})$ 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 we may write this as: $\forall p\in X-A\left[\neg\left(\exists U\in\mathcal{J}[p\in U\wedge U\subseteq A\right)\right]$ $\iff\forall p\in X-A\forall U\in\mathcal{J}[\neg(p\in U\wedge U\subseteq A)]$ $\iff\forall p\in X-A\forall U\in\mathcal{J}[(\neg(p\in U))\vee(\neg(U\subseteq A))]$ - by the negation of logical and $\iff\forall p\in X-A\forall U\in\mathcal{J}[p\notin U\vee U\nsubseteq A]$ - of course by the implies-subset relation we see $(A\subseteq B)\iff(\forall a\in A[a\in B])$, thus: $\iff\forall p\in X-A\forall U\in\mathcal{J}\big[p\notin U\vee(\exists q\in U[q\notin A])\big]$ TODO: Tidy this up
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