By Robert A. McCoy
This booklet brings jointly right into a normal atmosphere numerous strategies within the research of the topological houses of areas of constant capabilities. the 2 significant sessions of functionality house topologies studied are the set-open topologies and the uniform topologies. the place applicable, the analogous theorems for the 2 significant sessions of topologies are studied jointly, in order that a comparability may be made. A bankruptcy on cardinal features places characterizations of a couple of topological homes of functionality areas right into a extra common atmosphere: a few of these effects are new, others are generalizations of recognized theorems. Excercises are integrated on the finish of every bankruptcy, protecting different kinds of functionality area topologies. hence the publication will be applicable to be used in a school room environment in addition to for sensible research and basic topology. the single heritage wanted is a few simple wisdom of common topology.
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This ebook brings jointly right into a normal atmosphere numerous concepts within the examine of the topological houses of areas of constant capabilities. the 2 significant sessions of functionality area topologies studied are the set-open topologies and the uniform topologies. the place applicable, the analogous theorems for the 2 significant sessions of topologies are studied jointly, in order that a comparability may be made.
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Extra resources for Topological Properties of Spaces of Continuous Functions
Continuous. 3, the evaluation function e: E(f)(Z) ~ R is continuous. E(eo(E(f)xid)) -- E(f), so that f = eo(E(f)xid)is continuous. Therefore r But is a weakly conjoining topology. Finally suppose t h a t and let f • C(Z×X). r is a hypo-Ascoli topology, and let Z be a compact space Since Ep(f) is continuous, where Ep: C(ZxX) --* (Cp(X)) z is the exponential function, t h e n Ep(f)(Z) is a compact subset of Cp(X). 2 shows since r is hypo-Ascoli, E r ( f ) ( Z ) is compact. compact (Hausdorff) space is minimal Hausdorff, so that E r ( f ) ( Z ) = Ep(f)(Z).
For p a r t (b), it suffices to show for each f E C/t(XxY ) and each M e /a, t h a t 2 E ( ~ f ] ) = N'~E(f)]. Let g E fl, let (E(f)(x)(y),E(g)(x)(y)) x e E x M. and let Since y y E Y. is a r b i t r a r y , Then (f(x,y),g(x,y)) then Z M, (E(f)(x),E(g)(x)) 2 2 since x is a r b i t r a r y , (E(f),E(G)) E M, so t h a t E(g) E ME(f)]. so E ~1 that Also For the o t h e r inclusion, let 2 g ~ IVl[E(f)] and let (x,y) ~ X×Y. = (E(f)(x)(y)),g(x)(y)) E M. 7. ~[f], and thus g E E(Mf]). If Y is locally compact, then for • each space X, E: Ck(X×Y ) Ck(X,Ck(Y)) is a homeomorphism.
Now G can be characterized as being t o t a l l y m - b o u n d e d if and only if it is isomorphic to a subgroup of a group with cellularity less than or equal to m. Since Cp(X) has the countable chain condition, t h e n it is always t o t a l l y N0-bounded. 6. Proof. The space C a ( X ) is t o t a l l y m - b o u n d e d if and only if w a ( X ) __. m. Suppose f i r s t that C a ( X ) is t o t a l l y m-bounded. If A E a and if i: A $ X is the inclusion map, then the induced function homomorphism which is a continuous surjection.
Topological Properties of Spaces of Continuous Functions by Robert A. McCoy