Difference between revisions of "List of topological properties"

• $\overline{A}:\eq\bigcap\left\{C\in\mathcal{C}(X)\ \big\vert\ A\subseteq C\right\}$^[1] - where $\mathcal{C}(X)$ denotes the set of closed sets of $X$

Informally, it is the smallest closed set containing $A$.

• Note that the largest closed set c

• Interior
• Boundary

• $\forall U\in\mathcal{J}[U\cap A\neq\emptyset]$^{[1][Note 1]}

• $\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap A\neq\emptyset]$^[1]

Caveat:This is given as equiv to density by^[1] - also obviously follows from it!

• nowhere dense set

1. $\text{Closure}(A)$$\eq X$^[1]
2. $X-A$ contains no (non-empty) open subsets of $X$^[1]
   • Symbolically: $\forall U\in\mathcal{J}[U\nsubseteq X-A]$, which we can easily manipulate to get: $\forall U\in\mathcal{J}\exists p\in U[p\notin X-M]$
3. $X-A$ has no interior points^[1] (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:

     $\iff\forall p\in X-A\forall U\in\mathcal{J}\big[p\notin U\vee(\exists q\in U[q\notin A])\big]$

• $\exists U\in\mathcal{J}[a\in U\wedge U\subseteq A]$^[1]

• $\exists\epsilon>0[B_\epsilon(a)\subseteq A]$^[1]

Caveat:Basically follows from topological definition, these are closely related