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A construction of a Fréchet-Urysohn space, and some convergence concepts

Aleksander V. Arhangel'skii — 2010

Commentationes Mathematicae Universitatis Carolinae

Some strong versions of the Fréchet-Urysohn property are introduced and studied. We also strengthen the concept of countable tightness and generalize the notions of first-countability and countable base. A construction of a topological space is described which results, in particular, in a Tychonoff countable Fréchet-Urysohn space which is not first-countable at any point. It is shown that this space can be represented as the image of a countable metrizable space under a continuous pseudoopen mapping....

The Baire property in remainders of topological groups and other results

Aleksander V. Arhangel'skii — 2009

Commentationes Mathematicae Universitatis Carolinae

It is established that a remainder of a non-locally compact topological group G has the Baire property if and only if the space G is not Čech-complete. We also show that if G is a non-locally compact topological group of countable tightness, then either G is submetrizable, or G is the Čech-Stone remainder of an arbitrary remainder Y of G . It follows that if G and H are non-submetrizable topological groups of countable tightness such that some remainders of G and H are homeomorphic, then the spaces...

Two types of remainders of topological groups

Aleksander V. Arhangel'skii — 2008

Commentationes Mathematicae Universitatis Carolinae

We prove a Dichotomy Theorem: for each Hausdorff compactification b G of an arbitrary topological group G , the remainder b G G is either pseudocompact or Lindelöf. It follows that if a remainder of a topological group is paracompact or Dieudonne complete, then the remainder is Lindelöf, and the group is a paracompact p -space. This answers a question in A.V. Arhangel’skii, , Moscow Univ. Math. Bull. 54:3 (1999), 1–6. It is shown that every Tychonoff space can be embedded as a closed subspace in a pseudocompact...

G δ -modification of compacta and cardinal invariants

Aleksander V. Arhangel'skii — 2006

Commentationes Mathematicae Universitatis Carolinae

Given a space X , its G δ -subsets form a basis of a new space X ω , called the G δ -modification of X . We study how the assumption that the G δ -modification X ω is homogeneous influences properties of X . If X is first countable, then X ω is discrete and, hence, homogeneous. Thus, X ω is much more often homogeneous than X itself. We prove that if X is a compact Hausdorff space of countable tightness such that the G δ -modification of X is homogeneous, then the weight w ( X ) of X does not exceed 2 ω (Theorem 1). We also establish...

Perfect mappings in topological groups, cross-complementary subsets and quotients

Aleksander V. Arhangel'skii — 2003

Commentationes Mathematicae Universitatis Carolinae

The following general question is considered. Suppose that G is a topological group, and F , M are subspaces of G such that G = M F . Under these general assumptions, how are the properties of F and M related to the properties of G ? For example, it is observed that if M is closed metrizable and F is compact, then G is a paracompact p -space. Furthermore, if M is closed and first countable, F is a first countable compactum, and F M = G , then G is also metrizable. Several other results of this kind are obtained....

Addition theorems for dense subspaces

Aleksander V. Arhangel'skii — 2015

Commentationes Mathematicae Universitatis Carolinae

We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space X which is the union of two dense metrizable subspaces need not be a p -space. However, if a normal space X is the union of a finite family μ of dense subspaces each of which is metrizable by a complete metric, then X is also metrizable by...

A generalization of Čech-complete spaces and Lindelöf Σ -spaces

Aleksander V. Arhangel'skii — 2013

Commentationes Mathematicae Universitatis Carolinae

The class of s -spaces is studied in detail. It includes, in particular, all Čech-complete spaces, Lindelöf p -spaces, metrizable spaces with the weight 2 ω , but countable non-metrizable spaces and some metrizable spaces are not in it. It is shown that s -spaces are in a duality with Lindelöf Σ -spaces: X is an s -space if and only if some (every) remainder of X in a compactification is a Lindelöf Σ -space [Arhangel’skii A.V., Remainders of metrizable and close to metrizable spaces, Fund. Math. 220 (2013),...

Continuous images of Lindelöf p -groups, σ -compact groups, and related results

Aleksander V. Arhangel'skii — 2019

Commentationes Mathematicae Universitatis Carolinae

It is shown that there exists a σ -compact topological group which cannot be represented as a continuous image of a Lindelöf p -group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf p -groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space Y is a continuous image of a Lindelöf p -group, then there exists a covering γ of Y by dyadic compacta such that | γ | 2 ω . We also show that if a homogeneous compact space Y is a continuous...

On topological and algebraic structure of extremally disconnected semitopological groups

Aleksander V. Arhangel'skii — 2000

Commentationes Mathematicae Universitatis Carolinae

Starting with a very simple proof of Frol’ık’s theorem on homeomorphisms of extremally disconnected spaces, we show how this theorem implies a well known result of Malychin: that every extremally disconnected topological group contains an open and closed subgroup, consisting of elements of order 2 . We also apply Frol’ık’s theorem to obtain some further theorems on the structure of extremally disconnected topological groups and of semitopological groups with continuous inverse. In particular, every...

On a theorem of W.W. Comfort and K.A. Ross

Aleksander V. Arhangel'skii — 1999

Commentationes Mathematicae Universitatis Carolinae

A well known theorem of W.W. Comfort and K.A. Ross, stating that every pseudocompact group is C -embedded in its Weil completion [5] (which is a compact group), is extended to some new classes of topological groups, and the proofs are very transparent, short and elementary (the key role in the proofs belongs to Lemmas 1.1 and 4.1). In particular, we introduce a new notion of canonical uniform tightness of a topological group G and prove that every G δ -dense subspace Y of a topological group G , such...

Moscow spaces, Pestov-Tkačenko Problem, and C -embeddings

Aleksander V. Arhangel'skii — 2000

Commentationes Mathematicae Universitatis Carolinae

We show that there exists an Abelian topological group G such that the operations in G cannot be extended to the Dieudonné completion μ G of the space G in such a way that G becomes a topological subgroup of the topological group μ G . This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the...

On α -normal and β -normal spaces

Aleksander V. Arhangel'skiiLewis D. Ludwig — 2001

Commentationes Mathematicae Universitatis Carolinae

We define two natural normality type properties, α -normality and β -normality, and compare these notions to normality. A natural weakening of Jones Lemma immediately leads to generalizations of some important results on normal spaces. We observe that every β -normal, pseudocompact space is countably compact, and show that if X is a dense subspace of a product of metrizable spaces, then X is normal if and only if X is β -normal. All hereditarily separable spaces are α -normal. A space is normal if and...

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