Weak Hausdorff space


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]

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

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 Edit

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   is k-closed in  
    • A subset   is k-closed, if   is closed in   for each compact  
  3. Each compact subspace is closed and strongly locally compact.
    • A space is strongly locally compact if for each   and each (not necessarily open) neighborhood   of   there exists a compact neighborhood   of   such that  

Properties Edit

  • A k-Hausdorff space is weak Hausdorff. For if   is k-Hausdorff and   is a continuous map from a compact space   then   is compact, hence Hausdorff, hence closed.
  • A Hausdorff space is k-Hausdorff. For a space is Hausdorff if and only if the diagonal   is closed in   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.

Δ-Hausdorff spaces Edit

A Δ-Hausdorff space is a topological space where the image of every path is closed; that is, if whenever   is continuous then   is closed in   Every weak Hausdorff space is  -Hausdorff, and every  -Hausdorff space is a T1 space. A space is Δ-generated if its topology is the finest topology such that each map   from a topological  -simplex   to   is continuous.  -Hausdorff spaces are to  -generated spaces as weak Hausdorff spaces are to compactly generated spaces.

See also Edit

References Edit

  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.