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 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.

References

^ Hoffmann, Rudolf-E. (1979), "On weak Hausdorff spaces", Archiv der Mathematik, 32 (5): 487–504, doi:10.1007/BF01238530, MR 0547371.
^ J.P. May, A Concise Course in Algebraic Topology. (1999) University of Chicago Press ISBN 0-226-51183-9 (See chapter 5)
^ Strickland, Neil P. (2009). "The category of CGWH spaces" (PDF).
^ 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.

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[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, MR 0251719.

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