Topological Space

Definition

Let $X$ be a topological space

1. A neighbourhood of $x\in X$ is an open set $A$ with $x\in A$.
2. $X$ is Hausdorff if for $x\ne y\exists \text{neighbourhoods}A,B$ of $x$ and $y$ respectively, such that $A\cap B=\varnothing$.
3. $A\subset X$ is closed if $A^{\complement }$ is open.
4. The closure (denoted by a bar over the set) $\overline{Y}$ of a subset $Y\subset X$ is the intersection of all closed subsets of $X$ that contain $Y$.
5. $X$ is compact if every open cover has a finite subcover.
6. $X$ is locally compact if any $x\in X$ has a neighbourhood with compact closure.
7. $X$ is $\sigma$-compact if it is a countable union of compact subsets with respect to the relative topology, i.e. an open set of a subset $Z$ of $X$ is of the type $Z\cap A$ where $A$ is open in $X$.

Dense

Say $Y$ is dense in $X$ if $\overline{Y}=X$. If $Y$ is countable.

Separable

Say $\overline{Y}=X$, then we say that $X$ is separable.