A compactification problem of J. De Groot
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Metrization theorem
Compactification (mathematics)
In response to questions of ArhangelâskiÄ, we present examples of (1) $({\text {MA}} + \neg {\text {CH}})$ a symmetrizable space which is not metrizable but has a completely normal compactification and (2) $({\text {CH}})$ a symmetrizable space which is not metrizable but has a perfectly normal compactification. In the construction of (2), a technique is developed which can be used to obtain first countable compactifications of many interesting examples.
Compactification (mathematics)
Metrization theorem
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One point compactification is studied in the light of ideal of subsets of $\mathbb{N}$. $\mathcal{I}$-proper map is introduced and showed that a continuous map can be extended continuously to the one point $\mathcal{I}$-compactification if and only if the map is $\mathcal{I}$-proper. Shrinking condition(C) introduced in this article plays an important role to study various properties of $\mathcal{I}$-proper maps. It is seen that one point $\mathcal{I}$-compactification of a topological space may fail to be Hausdorff but a class $\{\mathcal{I}\}$ of ideals has been identified for which one point $\mathcal{I}$-compactification coincides with the one point compactification if it is metrizable.
Compactification (mathematics)
Metrization theorem
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For a metrizable topological space X it is well known that in general the Čech-Stone compactification β(X) or the Wallman compactification W(X) are not metrizable. To remedy this fact one can alternatively associate a point-set distance to the metric, a so called approach distance. It is known that in this setting both a Čech-Stone compactification β*(X) and a Wallman compactification W*(X) can be constructed in such a way that their approach distances induce the original approach distance of the metric on X [23], [24].The main goal in this paper is to formulate necessary and sufficient conditions for an approach space X such that the Čech-Stone compactification β*(X) and the Wallman compactification W*(X) are isomorphic, thus answering a question first raised in [24]. The first clue to reach this goal is to settle a question left open in [11], to formulate sufficient conditions for a compact approach space to be normal. In particular the result shows that the Čech-Stone compactification β*(X) of a uniform T2 space, is always normal. We prove that the Wallman compactification W*(X) is normal if and only if X is normal, and we produce an example showing that, unlike for topological spaces, in the approach setting normality of X is not sufficient for β*(X) and W*(X) to be isomorphic. We introduce a strengthening of the regularity condition on X, which we call ideal-regularity, and in our main theorem we conclude that X is ideal-regular, normal and T1 if and only if X is a uniform T1 approach space with β*(X) and W*(X) isomorphic. Classical topological results are recovered and implications for (quasi-)metric spaces are investigated.
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Metrization theorem
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A compact group with {\bf all dense subspaces} separable is metrizable. Inspired by this, we conjecture that a compact group with {\bf all dense subgroups} separable is metrizable. Positive answers are given here for two elementary cases, say, when the compact group is additionally assumed to be abelian or connected. However, a locally compact abelian group, even when it has an open compact subgroup, with all dense subgroups separable, may not be metrizable. At the end of this note, it is shown that a locally compact group with {\bf all subgroups} separable is metrizable. Our arguments are formalized in a more general form, namely, not restricted to the countable case.
Metrization theorem
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Metrization theorem
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Second-countable space
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It is known that the Stone-Cech compactification of a metrizable space X is approximated by the collection of Smirnov compactifications of X for all compatible metrics on X. If we confine ourselves to locally compact separable metrizable spaces, the corresponding statement holds for Higson compactifications. We investigate the smallest cardinality of a set D of compatible metrics on X such that the Stone-Cech compactification of X is approximated by Smirnov or Higson compactifications for all metrics in D. We prove that it is either the dominating number or 1 for a locally compact separable metrizable space.
Metrization theorem
Compactification (mathematics)
Cardinality (data modeling)
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Compactification (mathematics)
Metrization theorem
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In response to questions of ArhangelâskiÄ, we present examples of (1) $({\text {MA}} + \neg {\text {CH}})$ a symmetrizable space which is not metrizable but has a completely normal compactification and (2) $({\text {CH}})$ a symmetrizable space which is not metrizable but has a perfectly normal compactification. In the construction of (2), a technique is developed which can be used to obtain first countable compactifications of many interesting examples.
Compactification (mathematics)
Metrization theorem
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Answering a question of Isbell we show that there exists a rim-compact space X such that every compactification Y of X has dim(Y\X)≥ 1.
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