Mathematics

We answer a question concerning classes of complete topological rings where unconditionally convergent series have a special property.

Let X and Y be topological spaces. Let C be a path-connected closed set of X×Y . Suppose that C is locally direct product, that is, for any (a,b)∈X×Y , there exist an open set U of X , an open set V of Y , a subset I of U and a subset J of V such that (a,b)∈U×V and C∩(U×V)=I×J hold. Then, in this memo, we show that C is globally so, that is, there exist a subset A of X and a subset B of Y such that C=A×B holds. The proof is elementary. Here, we note that one might be able to think of a (perhaps, open) similar problem for a fiber product of locally trivial fiber spaces, not just for a direct product of topological spaces. In Appendix, we mentioned a simple example of a C([0,1];R) -manifold that cannot be embedded in the direct product (C([0,1];R) ) n as a C([0,1];R) -submanifold. In addition, we introduce the concept of topological 2-space, which is locally the direct product of topological spaces and an analog of homotopy category for topological 2-space. Finally, we raise a question on the existence of an R n -Morse function and the existence of an R n -immersion in a finite-dimensional R n -Euclidean space. Here, we note that the problem of defining the concept of an R n -handle body may also be considered.

We give a proof of the well-known fact that the category of nearness spaces is bireflective in the category of merotopic spaces which uses Zorn's Lemma instead of the usual construction by transfinite induction.

Each series ∑ ∞ n=1 a n of real positive terms gives rise to a topology on N={1,2,3,...} by declaring a proper subset A⊆N to be closed if ∑ n∈A a n <∞ . We explore the relationship between analytic properties of the series and topological properties on N . In particular, we show that, up to homeomorphism, |R| -many topologies are generated. We also find an uncountable family of examples { N α } α∈[0,1] with the property that for any α<β , there is a continuous bijection N β → N α , but the only continuous functions N α → N β are constant.

The main purpose of this manuscript is to provide a short proof of the metrizability of F -metric spaces introduced by Jleli and Samet in \cite[\, Jleli, M. and Samet, B., On a new generalization of metric spaces, J. Fixed Point Theory Appl. (2018) 20:128]{JS1}.

Every topological space has a Kolmogorov quotient that is obtained by identifying topologically indistinguishable points, that is, points that are contained in exactly the same open sets. In this survey, we look at the relationship between topological spaces and their Kolmogorov quotients. In most natural examples of spaces, the Kolmogorov quotient is homeomorphic to the original space. A non-trivial relationship occurs, for example, in the case of pseudometric spaces, where the Kolmogorov quotient is a metric space. We also look at the topological indistinguishability relation in the context of topological groups and uniform spaces.

In this survey we catalogue the many results of the past several decades concerning bounds on the cardinality of a topological space with homogeneous or homogeneous-like properties. These results include van Douwen's Theorem, which states |X|≤ 2 πw(X) if X is a power homogeneous Hausdorff space, and its improvements |X|≤d(X ) πχ(X) and |X|≤ 2 c(X)πχ(X) for spaces X with the same properties. We also discuss de la Vega's Theorem, which states that |X|≤ 2 t(X) if X is a homogeneous compactum, as well as its recent improvements and generalizations to other settings. This reference document also includes a table of strongest known cardinality bounds on spaces with homogeneous-like properties. The author has chosen to give some proofs if they exhibit typical or fundamental proof techniques. Finally, a few new results are given, notably (1) |X|≤d(X ) πnχ(X) if X is homogeneous and Hausdorff, and (2) |X|≤πχ(X ) c(X)qψ(X) if X is a regular homogeneous space. The invariant πnχ(X) , defined in this paper, has the property πnχ(X)≤πχ(X) and thus (1) improves the bound d(X ) πχ(X) for homogeneous Hausdorff spaces. The invariant qψ(X) has the properties qψ(X)≤πχ(X) and qψ(X)≤ ψ c (X) if X is Hausdorff, thus (2) improves the bound 2 c(X)πχ(X) in the regular, homogeneous setting.

A Hausdorff topological space X is called superconnected (resp. coregular ) if for any nonempty open sets U 1 ,… U n ⊆X , the intersection of their closures U ¯ 1 ∩⋯∩ U ¯ n is not empty (resp. the complement X∖( U ¯ 1 ∩⋯∩ U ¯ n ) is a regular topological space). A canonical example of a coregular superconnected space is the projective space Q P ∞ of the topological vector space Q <ω ={( x n ) n∈ω ∈ Q ω :|{n∈ω: x n ≠0}|<ω} over the field of rationals Q . The space Q P ∞ is the quotient space of Q <ω ∖{0 } ω by the equivalence relation x∼y iff Q⋅x=Q⋅y . We prove that every countable second-countable coregular space is homeomorphic to a subspace of Q P ∞ , and a topological space X is homeomorphic to Q P ∞ if and only if X is countable, second-countable, and admits a decreasing sequence of closed sets ( X n ) n∈ω such that (i) X 0 =X , ⋂ n∈ω X n =∅ , (ii) for every n∈ω and a nonempty open set U⊆ X n the closure U ¯ contains some set X m , and (iii) for every n∈ω the complement X∖ X n is a regular topological space. Using this topological characterization of Q P ∞ we find topological copies of the space Q P ∞ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.

We show that if κ≤ω and there exists a group topology without non-trivial convergent sequences on an Abelian group H such that H n is countably compact for each n<κ then there exists a topological group G such that G n is countably compact for each n<κ and G κ is not countably compact. If in addition H is torsion, then the result above holds for κ= ω 1 . Combining with other results in the literature, we show that: a) Assuming c incomparable selective ultrafilters, for each n∈ω , there exists a group topology on the free Abelian group G such that G n is countably compact and G n+1 is not countably compact. (It was already know for ω ). b) If κ∈ω∪{ω}∪{ ω 1 } , there exists in ZFC a topological group G such that G γ is countably compact for each cardinal γ<κ and G κ is not countably compact.

We give the construction of an infinite topological space with unusual properties. The space is regular, separable, and connected, but removing any nonempty open set leaves the remainder of the space totally disconnected (in fact, totally separated). The space is also strongly Choquet (in fact, satisfies an even stronger condition) and has a basis with nice properties. The construction utilizes Jensen's diamond principle ♢ .