Weak Hausdorff space

Separation axioms in topological spaces
Kolmogorov classification
T0	(Kolmogorov)
T1	(Fréchet)
T2	(Hausdorff)
T2½	(Urysohn)
completely T2	(completely Hausdorff)
T3	(regular Hausdorff)
T3½	(Tychonoff)
T4	(normal Hausdorff)
T5	(completely normal  Hausdorff)
T6	(perfectly normal  Hausdorff)
History

In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed. ^[1] In particular, every Hausdorff space is weak Hausdorff. As a separation property, it is stronger than T₁, which is equivalent to the statement that points are closed. Specifically, every weak Hausdorff space is a T₁ space. ^{[2] [3]}

The notion was introduced by M. C. McCord ^[4] to remedy an inconvenience of working with the category of Hausdorff spaces. It is often used in tandem with compactly generated spaces in algebraic topology. For that, see the category of compactly generated weak Hausdorff spaces.

k-Hausdorff spaces

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

• 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.
• A space is Hausdorff-compactly generated weak Hausdorff if and only if it is Hausdorff-compactly generated k-Hausdorff.
• To show that the coherent topology induced by compact Hausdorff subspaces preserves the compact Hausdorff subspaces and their subspace topology requires that the space is k-Hausdorff; weak Hausdorff is not enough. Hence k-Hausdorff can be seen as the more fundamental definition.

Δ-Hausdorff spaces

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.

See also

• 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
• Separation axiom  – Axioms in topology defining notions of "separation"

References

1. Hoffmann, Rudolf-E. (1979), "On weak Hausdorff spaces", Archiv der Mathematik, 32 (5): 487–504, doi:10.1007/BF01238530, MR   0547371 .
2. J.P. May, A Concise Course in Algebraic Topology . (1999) University of Chicago Press ISBN   0-226-51183-9 (See chapter 5)
3. Strickland, Neil P. (2009). "The category of CGWH spaces" (PDF).
4. McCord, M. C. (1969), "Classifying spaces and infinite symmetric products", Transactions of the American Mathematical Society, 146: 273–298, doi: 10.2307/1995173 , JSTOR   1995173, MR   0251719 .
5. Lawson, J; Madison, B (1974). "Quotients of k-semigroups". Semigroup Forum. 9: 1–18. doi:10.1007/BF02194829.