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The (dis)connectedness of products of Hausdorff spaces in the box topology

Vitalij A. Chatyrko (2021)

Commentationes Mathematicae Universitatis Carolinae

In this paper the following two propositions are proved: (a) If X α , α ∈ A , is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product □ α ∈ A X α can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If X α , α ∈ A , is an infinite system of Brown Hausdorff topological spaces then the box product □ α ∈ A X α is also Brown Hausdorff, and hence, it is connected. A space is Brown if...

The Freudenthal space for approximate systems of compacta and some applications.

Ivan Loncar (1995)

Publicacions Matemàtiques

In this paper we define a space σ(X) for approximate systems of compact spaces. The construction is due to H. Freudenthal for usual inverse sequences [4, p. 153–156]. We establish the following properties of this space: (1) The space σ(X) is a paracompact space, (2) Moreover, if X is an approximate sequence of compact (metric) spaces, then σ(X) is a compact (metric) space (Lemma 2.4). We give the following applications of the space σ(X): (3) If X is an approximate system of continua, then X = limX...

The functor σ²X

Stevo Todorčević (1995)

Studia Mathematica

We disprove the existence of a universal object in several classes of spaces including the class of weakly Lindelöf Banach spaces.

The G δ -topology and incompactness of ω α

Isaac Gorelic (1996)

Commentationes Mathematicae Universitatis Carolinae

We establish a relation between covering properties (e.gĿindelöf degree) of two standard topological spaces (Lemmas 4 and 5). Some cardinal inequalities follow as easy corollaries.

The hyperspace of finite subsets of a stratifiable space

Robert Cauty, Bao-Lin Guo, Katsuro Sakai (1995)

Fundamenta Mathematicae

It is shown that the hyperspace of non-empty finite subsets of a space X is an ANR (an AR) for stratifiable spaces if and only if X is a 2-hyper-locally-connected (and connected) stratifiable space.

The Lindelöf property and pseudo- ℵ 1 -compactness in spaces and topological groups

Constancio Hernández, Mihail G. Tkachenko (2008)

Commentationes Mathematicae Universitatis Carolinae

We introduce and study, following Z. Frol’ık, the class ℬ ( 𝒫 ) of regular P -spaces X such that the product X × Y is pseudo- ℵ 1 -compact, for every regular pseudo- ℵ 1 -compact P -space Y . We show that every pseudo- ℵ 1 -compact space which is locally ℬ ( 𝒫 ) is in ℬ ( 𝒫 ) and that every regular Lindelöf P -space belongs to ℬ ( 𝒫 ) . It is also proved that all pseudo- ℵ 1 -compact P -groups are in ℬ ( 𝒫 ) . The problem of characterization of subgroups of ℝ -factorizable (equivalently, pseudo- ℵ 1 -compact) P -groups is considered as well. We give some necessary...

The nonexistence of expansive homeomorphisms of chainable continua

Hisao Kato (1996)

Fundamenta Mathematicae

A homeomorphism f:X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x ≠ y, then there is an integer n ∈ ℤ such that d ( f n ( x ) , f n ( y ) ) > c . In this paper, we prove that if a homeomorphism f:X → X of a continuum X can be lifted to an onto map h:P → P of the pseudo-arc P, then f is not expansive. As a corollary, we prove that there are no expansive homeomorphisms on chainable continua. This is an affirmative answer to one of Williams’ conjectures.

The notion of closedness in topological categories

Mehmet Baran (1993)

Commentationes Mathematicae Universitatis Carolinae

In [1], various generalizations of the separation properties, the notion of closed and strongly closed points and subobjects of an object in an arbitrary topological category are given. In this paper, the relationship between various generalized separation properties as well as relationship between our separation properties and the known ones ([4], [5], [7], [9], [10], [14], [16]) are determined. Furthermore, the relationships between the notion of closedness and strongly closedness are investigated...

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