# Definition
Let $X$ be a topological space
1. A **neighbourhood** of $x \in X$ is an [[Open Sets|open set]] $A$ with $x \in A$.
2. $X$ is Hausdorff if for $x \neq y\; \exists \, \text{neighbourhoods} \, A, B$ of $x$ and $y$ respectively, such that $A \cap B = \emptyset$.
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|compact]]** if every [[Open Cover|open cover]] has a finite [[Subcover|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 Sets|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**.