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### 3-dimensional AR's which do not contain 2-dimensional ANR's

Fundamenta Mathematicae

### A 3-dimensional irreducible compact absolute retract which contains no disc

Fundamenta Mathematicae

### A Čech function in ZFC

Fundamenta Mathematicae

A nontrivial surjective Čech closure function is constructed in ZFC.

### A C(K) Banach space which does not have the Schroeder-Bernstein property

Studia Mathematica

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

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 $\left(X,{\tau }_{X}\right)$ and $\left(Y,{\tau }_{Y}\right)$ are called T₁-complementary provided that there exists a bijection f: X → Y such that ${\tau }_{X}$ and ${f}^{-1}\left(U\right):U\in {\tau }_{Y}$ 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

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

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

Commentationes Mathematicae Universitatis Carolinae

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

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.

International Journal of Mathematics and Mathematical Sciences

### A counter-example concerning quasi-homeomorphisms of compacta

Fundamenta Mathematicae

### A counter-example on unicoherent Peano spaces

Colloquium Mathematicae

### A counterexample to a theorem of Argyros

Matematički Vesnik

### A dimension raising hereditary shape equivalence

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

Acta Universitatis Carolinae. Mathematica et Physica

### A Dowker group

Commentationes Mathematicae Universitatis Carolinae

### A factorization theorem and its application to extremally disconnected resolutions

Colloquium Mathematicae

### A first countable supercompact Hausdorff space with a closed ${G}_{\delta }$ nonsupercompact subspace

Colloquium Mathematicae

### A forcing construction of thin-tall Boolean algebras

Fundamenta Mathematicae

It was proved by Juhász and Weiss that for every ordinal α with $0<\alpha <{\omega }_{2}$ there is a superatomic Boolean algebra of height α and width ω. We prove that if κ is an infinite cardinal such that ${\kappa }^{<\kappa }=\kappa$ and α is an ordinal such that $0<\alpha <{\kappa }^{++}$, 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 $\alpha <{\kappa }^{++}$, we obtain a notion of forcing that preserves cardinals and such that in the corresponding generic...

### A group topology on the free abelian group of cardinality 𝔠 that makes its square countably compact

Fundamenta Mathematicae

Under 𝔭 = 𝔠, we prove that it is possible to endow the free abelian group of cardinality 𝔠 with a group topology that makes its square countably compact. This answers a question posed by Madariaga-Garcia and Tomita and by Tkachenko. We also prove that there exists a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group) whose square is countably compact. This answers a question posed by Grant.

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