TY - JOUR

T1 - Function-space compactifications of function spaces

AU - Escardo, Martin

PY - 2002/5/15

Y1 - 2002/5/15

N2 - If X and Y are Hausdorff spaces with X locally compact, then the compact-open topology on the set C(X, Y) of continuous maps from X to Y is known to produce the right function-space topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for any Tychonoff space Y, there is a densely injective space Z containing Y as a densely embedded subspace such that, for every locally compact space X, the set C(X, Z) has a compact Hausdorff topology whose relative topology on C(X, Y) is the compact-open topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X, Y) under the compact-open topology is embedded into the Vietoris hyperspace V(X x Y). (2) The space of real-valued continuous functions on a locally compact Hausdorff space under the compact-open topology is embedded into a compact Hausdorff space whose points a-re pairs of extended real-valued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways. (C) 2002 Elsevier Science B.V. All rights reserved.

AB - If X and Y are Hausdorff spaces with X locally compact, then the compact-open topology on the set C(X, Y) of continuous maps from X to Y is known to produce the right function-space topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for any Tychonoff space Y, there is a densely injective space Z containing Y as a densely embedded subspace such that, for every locally compact space X, the set C(X, Z) has a compact Hausdorff topology whose relative topology on C(X, Y) is the compact-open topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X, Y) under the compact-open topology is embedded into the Vietoris hyperspace V(X x Y). (2) The space of real-valued continuous functions on a locally compact Hausdorff space under the compact-open topology is embedded into a compact Hausdorff space whose points a-re pairs of extended real-valued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways. (C) 2002 Elsevier Science B.V. All rights reserved.

UR - http://www.scopus.com/inward/record.url?scp=0038351004&partnerID=8YFLogxK

U2 - 10.1016/S0166-8641(01)00089-X

DO - 10.1016/S0166-8641(01)00089-X

M3 - Article

VL - 120

SP - 441

EP - 463

JO - Topology and its Applications

JF - Topology and its Applications

IS - 3

ER -