Weak Hausdorff space

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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 says that points are closed): 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.


  1. ^ Hoffmann, Rudolf-E. (1979), "On weak Hausdorff spaces", Archiv der Mathematik, 32 (5): 487–504, doi:10.1007/BF01238530, MR 547371 .
  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, MR 0251719 .