 T1 space

A topological space is T1 when every point is a closed point. Equivalently, for distinct $x,y$ there is an open set containing $x$ but not $y$, and an open set containing $y$ but not $x$. These open sets need not be disjoint. Every Hausdorff space is T1, but an infinite set with its cofinite topology is T1 without being Hausdorff.

In topology, a **T1 space** (also known as a **Fréchet space**) is a type of topological space that satisfies a particular separation axiom. Specifically, a topological space \( X \) is considered T1 if, for any two distinct points \( x \) and \( y \) in \( X \), there are open sets that separate these points.