Let X be a paracompact space of indX [greater than or equal to] n and let G be a finite group acting freely on X and H a cyclic subgroup of G of prime order p.
Let Xbe a paracompact space and let G be a finite group acting freely on X.
If X is a metacompact space or a subparacompact space and [mu] [member of] [M.sub.[tau]](X), then the subspace [supp.sub.X]([mu]) is Lindelof (, Theorem 27 for a paracompact space X).
A space X is called a uniformly Prohorov space if for each [epsilon] > 0 any paracompact space Z and any kernel k: Z [right arrow] [M.sub.r](X) there exists an upper semi-continuous compact-valued mapping [S.sub.(k,[epsilon])]: Z [right arrow] X such that [mu].sub.(k, z)(X\[S.sub.(k,[epsilon])](z)) [less than or equal to] [epsilon] for each z [member of] Z.
Consequently, it is a locally compact countable at infinity space and a paracompact space
, which admits the partition of unity by smooth real functions.
Let B be a paracompact space of finite paracompact dimension and [phi] : B [right arrow] Gr(H) a continuous map.
We conclude this section by stating that Gr(H) is a classifying space for Hilbert space bundles over paracompact spaces. This follows from the following three facts: First, any continuous field of infinite dimensional Hilbert spaces over a paracompact space B is isomorphic to the pullback over some map of the canonical field D over Gr(H).
Definition 2.6. A paracompact space
(X, [tau]) is a Hausdorff space with the property that every open cover of X has an open locally finite refiniment.
() Let f be a map and i: X [right arrow] Y a cofibration replacement for f with Y a paracompact space
. Then secat(f) is the smallest m such that there exists a map r making the following diagram homotopy commutative:
I (Existence) For every complex bundle [eta] over a finite dimensional paracompact space
B and every integer i [greater than or equal to] 0 there exists a Chern class [c.sub.i]([eta]) in [H.sup.2i](B; Z).
It is well-known that the minimal dense linearly ordered extension l(X) of a paracompact space
X may not be paracompact, however for the minimal dense linearly ordered extension l(M) of the Michael line M, we have the following Theorem.
Ramadan, 1991, On pairwise paracompact spaces