Much work has been done on topologies which are “determined” by sequences or more generally, a specified class of nets. The general concept of a space whose topology is “determined” by a given class of nets is studied. A unification and extension of the published work is obtained. In particular, product spaces are studied in detail.
It is proved that if X is a sequentially compact Hausdorff space, E a Hausdorff complete uniform space, C(X, E) the space of all E-valued continuous functions on X with uniformity, being the class of all compact subsets of X, and H a closed subset of C(X, E) containing a countable union of precompact subsets of C(X, E) as a dense subset, then H is complete.
The homotopy lifting property is not a very useful notion when applied to maps p:E→B between spaces with bad local properties. The approximate homotopy lifting property, introduced by D.S. Coram and P.F. Duvall, is useful only when E and B are ANR's. This paper introduces a new class of maps p:E→B between locally compact metric spaces called shape fibrations. Shape fibrations are defined in the spirit of the ANR-sequence approach to shape theory. It is shown that shape fibrations coincide with approximate fibrations whenever the base space and total space are ANR's. The following are typical results:
(i) fibers have the same shape whenever the base space is path connected,
(ii) any proper cell-like map between finite-dimensional locally compact metric spaces is a
shape fibration, and
(iii) the Taylor map is a cell-like map which fails to be a shape fibration.
Using the fact that each product of uniform quotient mappings is a quotient mapping, new conditions are given for the finite and countable productivity of a coreflective sub-class of uniform spaces. Three basis examples of productive coreflective sub-classes are constructed (connected with products of discrete spaces, proximally fine spaces, and uniformly sequentially continuous mappings) and the coreflective hull of metric spaces is shown to be productive if and only if there exists no uniformly sequential cardinal number.
Our discussion answers the questions as to what topological spaces are statistically metrizable in the sense of Schweizer and Sklar [5] and whether this can be discerned by the t-norm on the space in the Menger triangle relation. Namely we prove (1) the class of topological Menger spaces coincides with that of semi-metrizable topological spaces, and (2) no condition weaker than 1=supx<1T(x, x) can guarantee that a Menger space satisfying the Menger triangle relation under T is topological.
This paper offers a unified approach to a number of diverse results claiming the normality of certain spaces, under MA.
If X is a topological space with density d(X)⩾2, then cf (d((Xκ)(λ)))⩾cf λ, where (Xκ)(λ) is the λ-box product of κ copies of X. We use this observation to get lower bounds for the function δ(κ, λ)=d((D(2)κ)(λ)), where D(2) is the discrete space {0, 1}. It turns out that δ(κ, λ) is usually (if not always) equal to the well-known upper bound (log κ)<λ. We also answer a question of Confort and Negrepontis about necessary and sufficient conditions for δ(κ+, λ)⩽κ.
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