On Closed Images of the Space of Irrationals

Engelking, R.

doi:10.2307/2036424

Cited by 3 publications

(3 citation statements)

References 3 publications

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“…By Lemma 8.3 of [7], X contains as a closed subspace the countable metric fan M . Since X ′ is not compact, X contains also an isomorphic copy of N × s. Thus X contains a closed subspace Y which is homeomorphic to the space L. As Y is a retract of X (see [8] for a more general assertion) and X has the Dugundji extension property by [5], C k (Y, 2) is a retract of C k (X, 2). So C k (X, 2) is not Ascoli by Proposition 2.6 and [4, Proposition 5.2].…”

Section: Proposition 31 ([23]mentioning

confidence: 99%

Topological properties of function spaces C(X,2) over zero-dimensional metric spaces X

Gabriyelyan

2016

Topology and its Applications

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Let X be a zero-dimensional metric space and X ′ its derived set. We prove the following assertions: (1) the space C k (X, 2) is an Ascoli space iff C k (X, 2) is k R -space iff either X is locally compact or X is not locally compact but X ′ is compact, (2) C k (X, 2) is a k-space iff either X is a topological sum of a Polish locally compact space and a discrete space or X is not locally compact but X ′ is compact, (3) C k (X, 2) is a sequential space iff X is a Polish space and either X is locally compact or X is not locally compact but X ′ is compact, (4) C k (X, 2) is a Fréchet-Urysohn space iff C k (X, 2) is a Polish space iff X is a Polish locally compact space, (5) C k (X, 2) is normal iff X ′ is separable, (6) C k (X, 2) has countable tightness iff X is separable. In cases (1)-(3) we obtain also a topological and algebraical structure of C k (X, 2).

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Section: Proposition 31 ([23]mentioning

confidence: 99%

Topological properties of function spaces C(X,2) over zero-dimensional metric spaces X

Gabriyelyan

2016

Topology and its Applications

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“…This contradicts Proposition 7. If f is closed-resolvable, we use a continuous closed surjection g : ω ω → X given by the theorem of Engelking [1] (cf. proof of Proposition 3) and obtain a contradiction in a similar fashion.…”

Section: Main Theoremmentioning

confidence: 99%

Resolvable maps preserve complete metrizability

Gao

Kieftenbeld

2010

Proc. Amer. Math. Soc.

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“…"Only if" now follows from a theorem of Vaΐnsteΐn [16] asserting that every metrizable image of a complete metric space under a closed continuous map has a complete metric. "If" is a recent result of R. Engelking [4]; he shows, more generally, that every nonempty complete metric space of weight m is the image of B(m) under closed a continuous map.…”

Section: Yev N =F(x Nj Jc:f(u)mentioning

confidence: 99%