A k-Hausdorff space^[5] is a topological space which satisfies any of the following equivalent conditions:

1. Each compact subspace is Hausdorff.
2. The diagonal ${\displaystyle \{(x,x):x\in X\}}$  is k-closed in ${\displaystyle X\times X.}$ 
   • A subset ${\displaystyle A\subseteq Y}$  is k-closed, if ${\displaystyle A\cap C}$  is closed in ${\displaystyle C}$  for each compact ${\displaystyle C\subseteq Y.}$ 
3. Each compact subspace is closed and strongly locally compact.
   • A space is strongly locally compact if for each ${\displaystyle x\in X}$  and each (not necessarily open) neighborhood ${\displaystyle U\subseteq X}$  of ${\displaystyle x,}$  there exists a compact neighborhood ${\displaystyle V\subseteq X}$  of ${\displaystyle x}$  such that ${\displaystyle V\subseteq U.}$ 

Properties edit

• A k-Hausdorff space is weak Hausdorff. For if ${\displaystyle X}$  is k-Hausdorff and ${\displaystyle f:C\to X}$  is a continuous map from a compact space ${\displaystyle C,}$  then ${\displaystyle f(C)}$  is compact, hence Hausdorff, hence closed.
• A Hausdorff space is k-Hausdorff. For a space is Hausdorff if and only if the diagonal ${\displaystyle \{(x,x):x\in X\}}$  is closed in ${\displaystyle X\times X,}$  and each closed subset is a k-closed set.
• A k-Hausdorff space is KC. A KC space is a topological space in which every compact subspace is closed.
• To show that the coherent topology induced by compact Hausdorff subspaces preserves the compact Hausdorff subspaces and their subspace topology requires that the space be k-Hausdorff; weak Hausdorff is not enough. Hence k-Hausdorff can be seen as the more fundamental definition.

A Δ-Hausdorff space is a topological space where the image of every path is closed; that is, if whenever ${\displaystyle f:[0,1]\to X}$  is continuous then ${\displaystyle f([0,1])}$  is closed in ${\displaystyle X.}$  Every weak Hausdorff space is ${\displaystyle \Delta }$ -Hausdorff, and every ${\displaystyle \Delta }$ -Hausdorff space is a T₁ space. A space is Δ-generated if its topology is the finest topology such that each map ${\displaystyle f:\Delta ^{n}\to X}$  from a topological ${\displaystyle n}$ -simplex ${\displaystyle \Delta ^{n}}$  to ${\displaystyle X}$  is continuous. ${\displaystyle \Delta }$ -Hausdorff spaces are to ${\displaystyle \Delta }$ -generated spaces as weak Hausdorff spaces are to compactly generated spaces.

• Fixed-point space – Topological space such that every endomorphism has a fixed point, a Hausdorff space where every continuous function from the space into itself has a fixed point.
• Hausdorff space – Type of topological space
• Locally Hausdorff space
• Particular point topology
• Quasitopological space – a set X equipped with a function that associates to every compact Hausdorff space K a collection of maps K→C satisfying certain natural conditionsPages displaying wikidata descriptions as a fallback
• Separation axiom – Axioms in topology defining notions of "separation"