3-dimensional AR's which do not contain 2-dimensional ANR's
A 3-dimensional irreducible compact absolute retract which contains no disc
A Čech function in ZFC
A nontrivial surjective Čech closure function is constructed in ZFC.
A C(K) Banach space which does not have the Schroeder-Bernstein property
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
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 and are called T₁-complementary provided that there exists a bijection f: X → Y such that and are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size 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
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 countably cellular topological group all of whose countable subsets are closed need not be -factorizable
We construct a Hausdorff topological group such that is a precalibre of (hence, has countable cellularity), all countable subsets of are closed and -embedded in , but is not -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
We construct a space having the properties in the title, and with the same technique, a countably compact topological group which is not absolutely countably compact.
A counter example on common periodic points of functions.
A counter-example concerning quasi-homeomorphisms of compacta
A counter-example on unicoherent Peano spaces
A counterexample to a theorem of Argyros
A dimension raising hereditary shape equivalence
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
A Dowker group
A first countable supercompact Hausdorff space with a closed nonsupercompact subspace
A group topology on the free abelian group of cardinality 𝔠 that makes its square countably compact
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.
A hedgehog in a product
A hereditarily indecomposable non-metric Hausdorff continuum
A nondegenerate σ-discrete Moore space which is connected