We investigate the mathematics of a model of the human mind which has been proposed by the psychologist Jens Mammen. Mathematical realizations of this model consists of what the first author (A.T.) has called Mammen spaces, where a Mammen space is a triple $(U,\mathcal S,\mathcal C)$, where U is a non-empty set (“the universe”), $\mathcal S$ is a perfect Hausdorff topology on U, and $\mathcal C\subseteq \mathcal P(U)$ together with $\mathcal S$ satisfy certain axioms.

We refute a conjecture put forward by Hoffmann-Jørgensen, who conjectured that the existence of a “complete” Mammen space implies the Axiom of Choice, by showing that in the first Cohen model, in which ZF holds but AC fails, there is a complete Mammen space. We obtain this by proving that in the first Cohen model, every perfect topology can be extended to a maximal perfect topology.

On the other hand, we also show that if all sets are Lebesgue measurable, or all sets are Baire measurable, then there are no complete Mammen spaces with a countable universe.

Further, we investigate two new cardinal invariants $\mathfrak u_{M}$ and $\mathfrak u_{T}$ associated with complete Mammen spaces and maximal perfect topologies, and establish some basic inequalities that are provable in ZFC. Then we show $\mathfrak u_{M}=\mathfrak u_{T}=2^{\aleph _{0}}$ follows from Martin’s Axiom, and, contrastingly, we show that $\aleph _{1}=\mathfrak u_{M}=\mathfrak u_{T}<2^{\aleph _{0}}=\aleph _{2}$ in the Baumgartner–Laver model.

Finally, consequences for psychology are discussed.

Erdős space $\mathfrak {E}$ and complete Erdős space $\mathfrak {E}_{c}$ have been previously shown to have topological characterizations. In this paper, we provide a topological characterization of the topological space $\mathbb {Q}\times \mathfrak {E}_{c}$ , where $\mathbb {Q}$ is the space of rational numbers. As a corollary, we show that the Vietoris hyperspace of finite sets $\mathcal {F}(\mathfrak {E}_{c})$ is homeomorphic to $\mathbb {Q}\times \mathfrak {E}_{c}$ . We also characterize the factors of $\mathbb {Q}\times \mathfrak {E}_{c}$ . An interesting open question that is left open is whether $\sigma \mathfrak {E}_{c}^{\omega }$ , the $\sigma $ -product of countably many copies of $\mathfrak {E}_{c}$ , is homeomorphic to $\mathbb {Q}\times \mathfrak {E}_{c}$ .

In this paper we study connections between topological games such as Rothberger, Menger, and compact-open games, and we relate these games to properties involving covers by ${{G}_{\delta }}$ subsets. The results include the following: (1) If TWO has a winning strategy in theMenger game on a regular space $X$, then $X$ is an Alster space. (2) If TWO has a winning strategy in the Rothberger game on a topological space $X$, then the ${{G}_{\delta }}$-topology on $X$ is Lindelöf. (3) The Menger game and the compact-open game are (consistently) not dual.

Recently the class of almost-N-continuous functions between topological spaces has been defined. This paper continues the study of such functions, especially from the point of view of changing the topology on the codomain.

It is a question of Arhangel'skiĭ [1] (Problem 2) whether the identity ψ(G) = X(G) holds for every minimal Hausdorff topological group G = 〈G,u〉). (Here, as usual, ψ(G), the pseudocharacter of G, is the least cardinal number K for which there is such that and and x(G), the character of G,is the least cardinality of a local base at e for (〈G,u〉.) That 〈G, u〉 is minimal means that, if v is a Hausdorff topological group topology for G and v ⊂ u, then v = u.

In this paper, we give some conditions on G sufficient to ensure a positive response to Arhangel'skiï's question, and we offer an example which responds negatively to a question on minimal groups posed some years ago (cf. [6] (p. 107) and [4] (p. 259)).

The paper discusses some consequences of weak monotonicity for connected maps in relation to essential connectedness of a space. The first main result gives conditions under which the image by a connected map of an essentially connected space is essentially connected. The second is that, for a connected mapping of a connected, 1 .c. space to a WLOTS-wise and essentially connected space, w-monotonicity implies monotonicity. The remainder of the paper discusses continuity properties of connected, w-monotone mappings with WLOTS-wise and essentially connected range.

We extend the comprehensive treatment of k-spaces and sequential spaces provided by Franklin's refined notion of a natural cover to kR-spaces and sR-spaces. For this purpose, an apparently unstudied class of maps of topological spaces, the class of R-quotient maps, is introduced.

It is shown that in a dense-in-itself Hausdorff space if every set having a dense interior is open, then every compact set is finite.