Sentence examples for be a paracompact space from inspiring English sources

Let X be a paracompact space and Y be a Banach space.

Let ({mathcal{P}}) be the class of polyhedra, (Xin D {mathcal{P}})) be a paracompact space with ({mathcal{E}}(X neq0) and (Gamma :Xrightrightarrows X) be a u.s.c.s.c

Let X be a paracompact space and let Y be a non-empty closed convex subset in a Hausdorff locally convex topological vector space and with the property ( K ).

Corollary 3.4 Let X be a paracompact space, ( Y, D ) be an LD-metric space, and Z be a closed subset of X with dim X Z ≤ 0. If F : X ⊸ Y is an alsc multimap such that F ( x ) is a D -set for all x ∈ X ∖ Z, then for ϵ > 0, F has an ϵ-approximate selection.

Corollary 3.5 Let X be a paracompact space, ( Y, Γ ) be an LC-metric space, and Z be a closed subset of X with dim X Z ≤ 0. If F : X ⊸ Y is an alsc multimap such that F ( x ) is Γ-convex for all x ∈ X ∖ Z, then for ϵ > 0, F has an ϵ-approximate selection.

Let ({mathcal{P}}) be the class of polyhedra and let (Xin D {mathcal{P}})) be a paracompact space with a well-defined non-trivial Euler-Poincaré characteristic ({mathcal{E}}(X)).17 Given any open cover ω of X, then there exists a locally finite polyhedron P such that: (i) P ω-homotopy dominates X, and   (ii) ({mathcal{E}}(P)) is well defined and non-trivial.  .

Let X be a paracompact topological space, Y be a normed vector space and T : X → 2 Y be a correspondence having convex values.

Theorem 3.7 Let X be a paracompact topological space, ( Y, D ) be a D-space with a uniformity U, and F : X ⊸ Y be a multimap with D -set values such that F ( X ) is of generalized Zima type.

Theorem 3.10 Let X be a paracompact topological space, ( Y, D ) be an LD-metric space, and Z be a closed subset of X with dim X Z ≤ 0. If F, T : X ⊸ Y are two multimaps such that (1) F and T are topologically separated;   (2) T is upper semicontinuous; and   (3) F is an alsc multimap such that F ( x ) is a D -set for all x ∈ X ∖ Z.  .

Let X be a paracompact topological space, Y be a locally convex topological vector space and let T : X → 2 Y be a correspondence with convex values.

Proposition 3.1 Let X be a paracompact topological space and ℛ be a locally finite open covering of X, ( Y, D ) be a D-space, and η : R → Y be a function.