Definition
A topological space X is called paracompact if every open cover has a locally finite open refinement.
Property
- Every paracompact Hausdorff space is normal.
- Every locally compact, paracompact Hausdorff space is strongly paracompact.
- Every locally metrizable paracompact Hausdorff space is metrizable.
- Every locally Cech-complete paracompact Haudorff space is a Cech-complete space.
- Every Lindelof, countably paracompact space is paracompact.
- Every metrizable space is paracompact.
- More generally, Dowker characterized Hausdorff paracompact spaces as those spaces that have the property if U is an open cover of X then there exists a U-mapping f from X onto some metrizable space M. Here f is a U-mapping if f is continuous and there exists an open cover V of M such that f^{-1}(V) refines U. See D-paracompact, [Dowker1948] and [Nashef1991] for detail.
- A Hausdorff space X is paracompact if and only if for each compactum Y the product X×Y is normal.
- A regular T_1-space X is a paracompact if and only if each open cover has a locally finite.
- Every collectionwise normal metacompact T_1-space is paracompact.
- The following conditions are equivalent for a Hausdorff space X:
- the space X is paracompact;
- for each open cover of X there is a locally finite partition of unity subordinated to it;
- for each open cover of X there is a partition of unity subordinated to it.
Remark
- Hausdorff性を仮定する事もある。
- If you replace "locally finite" with "point-finite", you obtain metacompact (pointwise paracompact, weakly paracompact).
- If you replace "locally finite" with "star-finite", you obtain hypocompact (starparacompact, strongly paracompact).
- If you replace "locally finite" with "compact-finite", you obtain mesocompact.
- If you replace "open refinement" with "refinement", you obtain subparacompact (α-paracompact).
Reference
- 松島与三 多様体入門, 裳華房 (1965)
- Engelking, General Topology, Taylor & Francis (June 1977)
- C. H. Dowker, An extension of Alexandroff's mapping theorem, Bull. Am. Math. Soc. Vol.54, pp.386-391 (1948).
- Bassam Al-Nashef, Cover-developements and D-paracompact spaces, Indian J. pure appl. Math. 22(2), pp.135-141, Feb. 1991.