Function spaces

Compact covers and function spaces

“…Note that if X and Y are p -equivalent and X is a µ-space, then the spaces X and Y are also c -equivalent, see [5]. It is well known (by a combination of Milyutin's and Pestov's results, see [5,Theorem 3] There are many known results about preservation and non-preservation of various topological properties by t-equivalence and p -equivalence; see, e.g., [5], [6], [8], [24], [32]. For example, metrizability, local compactness, the countability of weight, normality and paracompactness are not p -invariant.…”

“…They also gave an alternative proof for separable metrizable spaces by using Christensen's Theorem (1 below), [8,Theorem 5.1,Theorem 5.3]. Later on, Valov proved, using the results in [30], that the answer to the Arhangel'skii's problem is positive forČech-complete spaces Y , see [32,Corollary 4.6]. The combination of the above facts yields: The property of being a completely metrizable space is preserved by the p -equivalence for spaces satisfying the first axiom of countability.…”

“…We give a different proof of the preservation of complete metrizability in the case of c -equivalent spaces X and Y where X is Polish and Y is first-countable, and discuss the non-separable case. Note however that from Valov's quite technical [32,Corollary 4.6] it follows that if X is completely metrizable, Y is paracompact first countable, and there is a continuous surjection from C c (X) onto C c (Y ), then Y is completely metrizable. Our approach is different and uses a property of C c (X) which is preserved by linear open maps and characterizes spaces X with a compact resolution swallowing compact sets (Theorem 14, Corollary 15).…”

“…A space Y is a wq-space [32] if for each y ∈ Y there is a sequence {U n : n ∈ ω} of open neighbourhoods of y such that if y n ∈ U n for each n ∈ ω then {y n : n ∈ ω} is functionally bounded in Y . It is well known (and is easy to verify) that every space of pointwise countable type is a wq-space; moreover, every wq-and µ-space is of pointwise countable type.…”

Le degré de Lindelöf est $l$-invariant

“…J. Baars [5] a montré que c'estégalement le cas pour la classe des espaces qui sont paracompacts et qui vérifient le premier axiome de dénombrabilité. Le résultat de J. Baars aétéétendu par V. Valov [11]à la classe des wq-espaces età celle des espaces µ-complets.…”