A two-point set is a subset of the plane which meets every line in exactly two points. By working in models of set theory other than ZFC, we demonstrate two new constructions of two-point sets. Our first construction shows that in ZFC + CH there exist two-point sets which are contained within the union of a countable collection of concentric circles. Our second construction shows that in certain models of ZF, we can show the existence of two-point sets without explicitly invoking the Axiom of Choice....

We give several topological/combinatorial conditions that, for a filter on ω, are equivalent to being a non-meager -filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager -filter. Here, we identify a filter with a subspace of $2^{\omega }$ through characteristic functions. Along the way, we generalize to non-meager -filters a result of Miller (1984) about -points, and we employ and give a new proof of results of Marciszewski (1998). We also employ a theorem...

In the present paper we introduce a convergence condition $\left({\Sigma }^{\text{'}}\right)$ and continue the study of “not distinguish” for various kinds of convergence of sequences of real functions on a topological space started in [2] and [3]. We compute cardinal invariants associated with introduced properties of spaces.

In this paper we extend the notion of quasinormal convergence via ideals and consider the notion of $ℐ$-quasinormal convergence. We then introduce the notion of $ℐQN\left(ℐwQN\right)$ space as a topological space in which every sequence of continuous real valued functions pointwise converging to $0$, is also $ℐ$-quasinormally convergent to $0$ (has a subsequence which is $ℐ$-quasinormally convergent to $0$) and make certain observations on those spaces.

Quasi $P$-spaces are defined to be those Tychonoff spaces $X$ such that each prime $z$-ideal of $C\left(X\right)$ is either minimal or maximal. This article is devoted to a systematic study of these spaces, which are an obvious generalization of $P$-spaces. The compact quasi $P$-spaces are characterized as the compact spaces which are scattered and of Cantor-Bendixson index no greater than 2. A thorough account of locally compact quasi $P$-spaces is given. If $X$ is a cozero-complemented space and every nowhere dense zeroset...

Motivated by some examples, we introduce the concept of special almost P-space and show, using the reflection principle, that for every space $X$ of this kind the inequality “$\left|X\right|\le {\psi }_c{\left(X\right)}^{t\left(X\right)}$" holds.

Hewitt [Rings of real-valued continuous functions. I., Trans. Amer. Math. Soc. 64 (1948), 45–99] defined the $m$-topology on $C\left(X\right)$, denoted $C_m\left(X\right)$, and demonstrated that certain topological properties of $X$ could be characterized by certain topological properties of $C_m\left(X\right)$. For example, he showed that $X$ is pseudocompact if and only if $C_m\left(X\right)$ is a metrizable space; in this case the $m$-topology is precisely the topology of uniform convergence. What is interesting with regards to the $m$-topology is that it is possible, with...

L'operazione «anti( )» di Paul Bankston fu introdotta in contesto della famiglia di tutti gli spazii topologici. Però, per molte ricerche ci conviene lavorare esclusivamente in una classe costretta di spazii di cui la struttura e ricca abbastanza di facilitare il ragionamento. In quest'articolo descriviamo come trasferire anti ( ), e concetti allacciati, dentro una tale classe costretta; con riferimento speciale all'esistenza di «pre-antis».