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locally compact

Last-modified: 2010-12-04 (土) 16:27:49

Definition Edit

  • A topological space X is called locally compact if every point of X has a compact neighbourhood.

Property Edit

  • Every locally compact Hausdorff space is Tikhonov?.
  • Every locally compact, paracompact Hausdorff space is strongly paracompact.
  • For every compact subspace A of a locally compact space X and every open set V that contains A there exists an open set U such that imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20A%5csubset%20U%5csubset%20%5coverline%7bU%7d%5csubset%20V%20and%5c%20%5coverline%7bU%7d%20%5c%5d%7d%25.png is compact.
  • If X is locally compact space, then every subspace of X that can be represented in the form F∩V, where F is closed in X and V is open in X, also is locally compact.
  • Every locally compact subspace M of a Hausdorff space X is an open subset of the closure of the set M in the space X, i.e., it can be represented in the form F∩V, where F is closed in X and V is open in X.
  • The sum imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cbigoplus%7b%7d_%7bs%5cin%20S%7d%20X_s%20%5c%5d%7d%25.png is locally compact if and only if all spaces X_s are locally compact.
  • The Cartesian product imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5cprod%7b%7d_%7bs%5cin%20S%7d%20X_s%20%5c%5d%7d%25.png , where imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20X_s%5cneq%5cemptyset%5c%20%5ctext%7bfor%7d%5c%20s%5cin%20S%20%5c%5d%7d%25.png, is locally compact if and only if all spaces X_s are locally compact and there exists a finite set imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20S_0%5csubset%20S%20%5c%5d%7d%25.png such that X_s is compact for imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20s%5cin%20S%5cbackslash%20S_0%20%5c%5d%7d%25.png.
  • If there exists on open mapping f:X→Y of a locally compact space X onto a Hausdorff space Y, then Y is a locally compact space.
  • For every locally compact space X and any quotient mapping g:Y→Z, the Cartesian product f=Id_X×g;X×Y→X×Z is a quotient mapping.
  • If X is locally compact, then for every topological space Y the compact-open topology on Y^X is acceptable.
  • Every locally compact paracompact space X can be represented as the union of a family of disjoint closed-and-open subspaces of X each of which has the Lindeloef property.
  • Every non-empty hereditarily disconnected locally compact space is zero-dimensional.
  • Hereditary disconnectedness, zero-dimensionality and strong zero-dimensionality are equivalent in the realm of non-empty locally compact paracompact spaces.
  • For every non-emty locally compact paracompact space X the conditions ind X=0, Ind X=0 and dim X=0 are equivalent to hereditary disconnectedness of X.
  • a coarsest uniformity? on a Tykhonov? space X exists if and only if the space X is locally compact.
  • Every locally compact preregular space is completely regular.
  • Every locally compact Hausdorff space is a Baire space.
  • A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X can be written as the set-theoretic difference of two closed subsets of Y.

Reference Edit

John L. Kelley, General Topology, Springer (1975).