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We construct a totally disconnected compact Hausdorff space K₊ which has clopen subsets K₊” ⊆ K₊’ ⊆ K₊ such that K₊” is homeomorphic to K₊ and hence C(K₊”) is isometric as a Banach space to C(K₊) but C(K₊’) is not isomorphic to C(K₊). This gives two nonisomorphic Banach spaces (necessarily nonseparable) of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces...

A compact Hausdorff topology that is a T₁-complement of itself

Dmitri Shakhmatov, Michael Tkachenko (2002)

Fundamenta Mathematicae

Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X, τ_{X})$ and $(Y, τ_{Y})$ are called T₁-complementary provided that there exists a bijection f: X → Y such that $τ_{X}$ and $f^{- 1} (U) : U \in τ_{Y}$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size $2^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff...

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....

A construction of noncontractible simply connected cell-like two-dimensional Peano continua

Katsuya Eda, Umed H. Karimov, Dušan Repovš (2007)

Fundamenta Mathematicae

Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts from a noncontractible n-dimensional Peano continuum for any n > 0, then our construction yields a simply connected noncontractible (n + 1)-dimensional cell-like Peano continuum. In particular, starting from the circle 𝕊¹,...

A construction of the Gleason space

Aleksander Błaszczyk (1983)

Commentationes Mathematicae Universitatis Carolinae

A countably cellular topological group all of whose countable subsets are closed need not be $ℝ$ -factorizable

Mihail G. Tkachenko (2023)

Commentationes Mathematicae Universitatis Carolinae

We construct a Hausdorff topological group $G$ such that $ℵ_{1}$ is a precalibre of $G$ (hence, $G$ has countable cellularity), all countable subsets of $G$ are closed and $C$ -embedded in $G$ , but $G$ is not $ℝ$ -factorizable. This solves Problem 8.6.3 from the book “Topological Groups and Related Structures" (2008) in the negative.

A countably compact, separable space which is not absolutely countably compact

Jerry E. Vaughan (1995)

Commentationes Mathematicae Universitatis Carolinae

We construct a space having the properties in the title, and with the same technique, a countably compact $T_{2}$ topological group which is not absolutely countably compact.

A counter example on common periodic points of functions.

Alikhani-Koopaei, Aliasghar (1998)

International Journal of Mathematics and Mathematical Sciences

A counter-example concerning quasi-homeomorphisms of compacta

H. Patkowska (1978)

Fundamenta Mathematicae

A counter-example on unicoherent Peano spaces

J. H. V. Hunt (1971)

Colloquium Mathematicae

A counterexample to a theorem of Argyros

F. Richman (1990)

Matematički Vesnik

A dimension raising hereditary shape equivalence

Jan Dijkstra (1996)

Fundamenta Mathematicae

We construct a hereditary shape equivalence that raises transfinite inductive dimension from ω to ω+1. This shows that ind and Ind do not admit a geometric characterisation in the spirit of Alexandroff's Essential Mapping Theorem, answering a question asked by R. Pol.

A discontinuous function with a connected closed graph

Jiří Jelínek (2003)

Acta Universitatis Carolinae. Mathematica et Physica

A Dowker group

Klaas Pieter Hart, Heikki J. K. Junnila, Jan van Mill (1985)

Commentationes Mathematicae Universitatis Carolinae

A factorization theorem and its application to extremally disconnected resolutions

A. Błaszczyk (1973)

Colloquium Mathematicae

A first countable supercompact Hausdorff space with a closed $G_{δ}$ nonsupercompact subspace

Murray G. Bell (1980)

Colloquium Mathematicae

A forcing construction of thin-tall Boolean algebras

Juan Martínez (1999)

Fundamenta Mathematicae

It was proved by Juhász and Weiss that for every ordinal α with $0 < α < ω_{2}$ there is a superatomic Boolean algebra of height α and width ω. We prove that if κ is an infinite cardinal such that $κ^{< κ} = κ$ and α is an ordinal such that $0 < α < κ^{+ +}$ , then there is a cardinal-preserving partial order that forces the existence of a superatomic Boolean algebra of height α and width κ. Furthermore, iterating this forcing through all $α < κ^{+ +}$ , we obtain a notion of forcing that preserves cardinals and such that in the corresponding generic...

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