Separation Axioms

A space where for all $x, y \in X$, there exists $U$ such that $x \in U$ yet $y \not \in U$ OR vise versa. Also called distinguishable or T0. You might think these are useless, but notably, any topological space can be converted into a T0 space by factoring out indistinguishable points.

A space where for all $x, y \in X$ there exists $U, V$ such that $x \in U, y \in V$ yet $x \not \in V$, $y \not \in U$. These spaces are interesting because singletons are closed. For example take any singleton $\lbrace x \rbrace$ and consider the open set $\cup_{y \not = x} U_{y}$ where each open neighborhood of $y$ $U_{y}$ does not contain $x$. The complement of this set is closed, and is precisely the singleton.

A space where for all $x, y \in X$, there exists $U$, $V$ such that $x \in U$, $y \in V$, yet $U \cap V = \emptyset$. Notably limits on nets converge uniquely when they converge in these Hausdorff spaces.

A space where for all $x \in X$ and closed sets $F \subset X$ such that $x \not \in X$, there are open sets separating $F$ and $x$ in the same sense that they separate points in the Hausdorff spaces. Yet, it is possible for regular spaces under this definition to be not strictly stronger than Hausdorff spaces. For instance, not all singletons are closed in any topology. Therefore in order to restore the total ordering in terms of separation axiom strength, most people also define regular spaces to have to be Hausdorff Spaces as well. From here on out we will in general assume that these spaces are Hausdorff.

A space where for all $x \in X$ and closed sets $F \subset X$ such that $x \not \in X$, there is a continuous function $f: X \rightarrow [0, 1]$ that separates $x$ and $F$ such that $f(x) = 0$ and $f(F) \equiv 1$ (every point in $F$ maps to $1$). this property is interesting because of its theoretical importance in the Stone-Cech Compactification. Also called completely regular.

A space where for all closed $F, G \subset X$, there exists open sets $U, V$ separating them. This property is useful for applying Urysohn's Lemma.