Definition

Let X be a topological space

A neighbourhood of x \in X is an Open Sets A with x \in A.
X is Hausdorff if for x \neq y\; \exists \, \text{neighbourhoods} \, A, B of x and y respectively, such that A \cap B = \emptyset.
A \subset X is closed if A^{\complement} is open.
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.
X is Compact if every Open Cover has a finite Subcover.
X is locally compact if any x \in X has a neighbourhood with compact closure.
X is $\sigma$-compact if it is a countable union of compact subsets with respect to the relative topology, i.e. an Open Sets 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.

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