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A proof for the Blair-Hager-Johnson theorem on absolute z -embedding

Kaori Yamazaki (2002)

Commentationes Mathematicae Universitatis Carolinae

In this paper, a simple proof is given for the following theorem due to Blair [7], Blair-Hager [8] and Hager-Johnson [12]: A Tychonoff space X is z -embedded in every larger Tychonoff space if and only if X is almost compact or Lindelöf. We also give a simple proof of a recent theorem of Bella-Yaschenko [6] on absolute embeddings.

An example related to strongly pointwise self-homeomorphic dendrites

Pavel Pyrih (1999)

Archivum Mathematicum

Such spaces in which a homeomorphic image of the whole space can be found in every open set are called self-homeomorphic. W.J. Charatonik and A. Dilks posed a problem related to strongly pointwise self-homeomorphic dendrites. We solve this problem negatively in Example 2.1.

An independency result in connectification theory

Alessandro Fedeli, Attilio Le Donne (1999)

Commentationes Mathematicae Universitatis Carolinae

A space is called connectifiable if it can be densely embedded in a connected Hausdorff space. Let ψ be the following statement: “a perfect T 3 -space X with no more than 2 𝔠 clopen subsets is connectifiable if and only if no proper nonempty clopen subset of X is feebly compact". In this note we show that neither ψ nor ¬ ψ is provable in ZFC.

An observation on Krull and derived dimensions of some topological lattices

M. Rostami, Ilda I. Rodrigues (2011)

Archivum Mathematicum

Let ( L , ) , be an algebraic lattice. It is well-known that ( L , ) with its topological structure is topologically scattered if and only if ( L , ) is ordered scattered with respect to its algebraic structure. In this note we prove that, if L is a distributive algebraic lattice in which every element is the infimum of finitely many primes, then L has Krull-dimension if and only if L has derived dimension. We also prove the same result for error L , the set of all prime elements of L . Hence the dimensions on the lattice...

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...

Connectedness and local connectedness of topological groups and extensions

Ofelia Teresa Alas, Mihail G. Tkachenko, Vladimir Vladimirovich Tkachuk, Richard Gordon Wilson (1999)

Commentationes Mathematicae Universitatis Carolinae

It is shown that both the free topological group F ( X ) and the free Abelian topological group A ( X ) on a connected locally connected space X are locally connected. For the Graev’s modification of the groups F ( X ) and A ( X ) , the corresponding result is more symmetric: the groups F Γ ( X ) and A Γ ( X ) are connected and locally connected if X is. However, the free (Abelian) totally bounded group F T B ( X ) (resp., A T B ( X ) ) is not locally connected no matter how “good” a space X is. The above results imply that every non-trivial continuous homomorphism...

Diagonals and discrete subsets of squares

Dennis Burke, Vladimir Vladimirovich Tkachuk (2013)

Commentationes Mathematicae Universitatis Carolinae

In 2008 Juhász and Szentmiklóssy established that for every compact space X there exists a discrete D X × X with | D | = d ( X ) . We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf Σ -space X and hence X ω is d -separable. We give an example of a countably compact space X such that X ω is not d -separable. On the other hand, we show that for any Lindelöf p -space X there exists a discrete subset D X × X such that Δ = { ( x , x ) : x X } D ¯ ; in particular, the diagonal Δ is a retract of D ¯ and the projection...

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