Weak Hausdorff space

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 ${\displaystyle V\subseteq U.}$

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"