Mathematics

We study some variations of the product topology on families of clopen subsets of 2 N ×N in order to construct countable nodec regular spaces (i.e. in which every nowhere dense set is closed) with analytic topology which in addition are not selectively separable and do not satisfy the combinatorial principle q + .

In the paper, we investigate (scattered) compact spaces with a P -base for some poset P . More specifically, we prove that, under the assumption ω 1 <b , any compact space with an ω ω -base is first-countable and any scattered compact space with an ω ω -base is countable. These give positive solutions to Problems 8.6.9 and 8.7.7 in \cite{Banakh2019}. Using forcing, we also prove that in a model of ω 1 <b , there is a non-first-countable compact space with a P -base for some poset P with calibre~ ω 1 .

In this paper, we continue to study one of the classic problems in general topology raised by P.S. Alexandrov: when a Hausdorff space X has a continuous bijection (a condensation) onto a compactum? We concentrate on the situation when not only X but also X∖Y can be condensed onto a compactum whenever the cardinality of Y does not exceed certain τ .

We show that if X is a separable locally compact Hausdorff connected space with fewer than c non-cut points, then X embeds into a dendrite D⊆ R 2 , and the set of non-cut points of X is a nowhere dense G δ -set. We then prove a Tychonoff cut-point space X is weakly orderable if and only if βX is an irreducible continuum. Finally, we show every separable metrizable cut-point space densely embeds into a reducible continuum with no cut points. By contrast, there is a Tychonoff cut-point space each of whose compactifications has the same cut point. The example raises some questions about persistent cut points in Tychonoff spaces.

Let S be a topological property of sequences (such as, for example, "to contain a convergent subsequence" or "to have an accumulation point"). We introduce the following open-point game OP(X,S) on a topological space X. In the n'th move, Player A chooses a non-empty open subet U_n of X, and Player B responds by selecting a point x_n in U_n. Player B wins the game if the sequence (x_n) satisfies property S in X; otherwise, Player A wins. The (non-)existence of regular or stationary winning strategies in OP(X,S) for both players defines new compactness properties of the underlying space X. We thoroughly investigate these properties and construct examples distinguishing half of them, for an arbitrary property S sandwiched between sequential compactness and countable compactness.

The interrelations between various classes of convergence spaces defined by countability conditions are studied. Remarkably, they all find characterizations in the usual space of ultrafilters in terms of classical topological properties. This is exploited to produce relevant examples in the realm of convergence spaces from known topological examples.

We provide conceptual proofs of the two most fundamental theorems concerning topological games and open covers: Hurewicz's Theorem concerning the Menger game, and Pawlikowski's Theorem concerning the Rothberger game.

In this paper, we consider inverse limits of [0,1] using upper semicontinuous set-valued functions. We introduce two generalizations of the Intermediate Value Property and prove that inverse limits with upper semicontinuous set-valued bonding functions are connected if the bonding functions are surjective, have connected graphs, and have either generalization of the Intermediate Value Property. Examples are given to demonstrate that if any of the conditions is dropped, the result does not hold in general. An example is given to show that an inverse limit may be connected even if the bonding functions do not have either Intermediate Value Property. Further, we compare the structures of set-valued functions with the two types of the Intermediate Value Property.

The connected door space is an enigmatic topological space in which every proper nonempty subset is either open or closed, but not both. This paper provides an elementary proof of the classification theorem of connected door spaces. More importantly, we show that connected door topologies can be viewed as solutions of the valuation f(A)+f(B)=f(A∪B)+f(A∩B) and the equation f(A)+f(B)=f(A∪B) , respectively. In addition, some special solutions, which can be regarded as a union of connected door spaces, are provided.

It is obtained a natural generalisation of Uspenskij's selection characterisation of paracompact C -spaces. The method developed to achieve this result is also applied to give a simplified proof of a similar characterisation of paracompact finite C -space obtained previously by Valov. Another application is a characterisation of finite-dimensional paracompact spaces which generalises both a remark done by Michael and a result obtained by the author.