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]
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.
k-Hausdorff spaces
A k-Hausdorff space[5] is a topological space which satisfies any of the following equivalent conditions:
- Each compact subspace is Hausdorff.
- The diagonal is k-closed in .
- Each compact subspace is closed and strongly locally compact.
In these characterizations:
- A subset is k-closed, if is closed in for each compact .
- A space is strongly locally compact, if for each and a (not necessarily open) neighborhood of , there exists a compact neighborhood of such that .
Properties
- 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 k-closed.
- A k-Hausdorff space is KC. A space is KC, if each 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; i.e. if is continuous, then is closed. Every weak Hausdorff space is Δ-Hausdorff, and every Δ-Hausdorff space is T1. A space is Δ-generated, if its topology is the finest such that each map from a topological n-simplex to is continuous. Δ-Hausdorff spaces are to Δ-generated spaces as weak Hausdorff spaces are to compactly generated spaces.
See also
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, JSTOR 1995173, MR 0251719.
- Lawson, J; Madison, B (1974). "Quotients of k-semigroups". Semigroup Forum. 9: 1--18.