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Closed discrete subsets of separable spaces and relative versions of normality, countable paracompactness and property ( a )

Samuel Gomes da Silva (2011)

Commentationes Mathematicae Universitatis Carolinae

In this paper we show that a separable space cannot include closed discrete subsets which have the cardinality of the continuum and satisfy relative versions of any of the following topological properties: normality, countable paracompactness and property ( a ) . It follows that it is consistent that closed discrete subsets of a separable space X which are also relatively normal (relatively countably paracompact, relatively ( a ) ) in X are necessarily countable. There are, however, consistent examples of...

Closed embeddings into complements of Σ -products

Aleksander V. Arhangel'skii, Miroslav Hušek (2008)

Commentationes Mathematicae Universitatis Carolinae

In some sense, a dual property to that of Valdivia compact is considered, namely the property to be embedded as a closed subspace into a complement of a Σ -subproduct of a Tikhonov cube. All locally compact spaces are co-Valdivia spaces (and only those among metrizable spaces or spaces having countable type). There are paracompact non-locally compact co-Valdivia spaces. A possibly new type of ultrafilters lying in between P-ultrafilters and weak P-ultrafilters is introduced. Under Martin axiom and...

Closed graph multi-selections

Valentin Gutev (2011)

Fundamenta Mathematicae

A classical Lefschetz result about point-finite open covers of normal spaces is generalised by showing that every lower semi-continuous mapping from a normal space into the nonempty compact subsets of a metrizable space admits a closed graph multi-selection. Several applications are given as well.

Closed ideals in topological algebras: a characterization of the topological Φ -algebra C k ( X )

F. Montalvo, Antonio A. Pulgarín, Batildo Requejo Fernández (2006)

Czechoslovak Mathematical Journal

Let A be a uniformly closed and locally m-convex Φ -algebra. We obtain internal conditions on A stated in terms of its closed ideals for A to be isomorphic and homeomorphic to C k ( X ) , the Φ -algebra of all the real continuous functions on a normal topological space X endowed with the compact convergence topology.

Closed mapping theorems on k -spaces with point-countable k -networks

Alexander Shibakov (1995)

Commentationes Mathematicae Universitatis Carolinae

We prove some closed mapping theorems on k -spaces with point-countable k -networks. One of them generalizes Lašnev’s theorem. We also construct an example of a Hausdorff space U r with a countable base that admits a closed map onto metric space which is not compact-covering. Another our result says that a k -space X with a point-countable k -network admitting a closed surjection which is not compact-covering contains a closed copy of U r .

Closed subgroups in Banach spaces

Fredric Ancel, Tadeusz Dobrowolski, Janusz Grabowski (1994)

Studia Mathematica

We show that zero-dimensional nondiscrete closed subgroups do exist in Banach spaces E. This happens exactly when E contains an isomorphic copy of c 0 . Other results on subgroups of linear spaces are obtained.

Closed subsets of absolutely star-Lindelöf spaces II

Yan-Kui Song (2003)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we prove the following two statements: (1) There exists a discretely absolutely star-Lindelöf Tychonoff space having a regular-closed subspace which is not CCC-Lindelöf. (2) Every Hausdorff (regular, Tychonoff) linked-Lindelöf space can be represented in a Hausdorff (regular, Tychonoff) absolutely star-Lindelöf space as a closed G δ subspace.

Closure spaces and characterizations of filters in terms of their Stone images

Anh Tran Mynard, Frédéric Mynard (2007)

Czechoslovak Mathematical Journal

Fréchet, strongly Fréchet, productively Fréchet, weakly bisequential and bisequential filters (i.e., neighborhood filters in spaces of the same name) are characterized in a unified manner in terms of their images in the Stone space of ultrafilters. These characterizations involve closure structures on the set of ultrafilters. The case of productively Fréchet filters answers a question of S. Dolecki and turns out to be the only one involving a non topological closure structure.

Closure-preserving covers in function spaces

David Guerrero Sánchez (2010)

Commentationes Mathematicae Universitatis Carolinae

It is shown that if C p ( X ) admits a closure-preserving cover by closed σ -compact sets then X is finite. If X is compact and C p ( X ) has a closure-preserving cover by separable subspaces then X is metrizable. We also prove that if C p ( X , [ 0 , 1 ] ) has a closure-preserving cover by compact sets, then X is discrete.

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