Related papers: $\mathfrak P_0$-spaces

A Hausdorff topological group is called minimal if it does not admit a strictly coarser Hausdorff group topology. This paper mostly deals with the topological group $H_+(X)$ of order-preserving homeomorphisms of a compact linearly ordered…

A space $X$ is called a generalized Namioka space (g$\mathcal{N}$-space), if for every compact space $Y$ and every separately continuous function $f\colon X\times Y\rightarrow\mathbb{R}$, there exists at least one point $x\in X$ such that…

The enumeration degrees of sets of natural numbers can be identified with the degrees of difficulty of enumerating neighborhood bases of points in a universal second-countable $T_0$-space (e.g. the $\omega$-power of the Sierpi\'nski space).…

We classify all continuous tensor product systems of Hilbert spaces which are ``infinitely divisible" in the sense that they have an associated logarithmic structure. These results are applied to the theory of E_0 semigroups to deduce that…

A Lambda (Card Lambda > aleph)-product space of {0,1} has a partition {X1,...,Xn} for any n a decomposition space of each Xi of which is self-similar.(February 16, 2015)

We give a characterization of countable discrete subspace $A$ of a topological space $X$ such that there exists a (linear) continuous mapping $\varphi:C_p^*(A)\to C_p(X)$ with $\varphi(y)|_A=y$ for every $y\in C_p^*(A)$. Using this…

A topological space is called a submetrizable if it can be mapped onto a metrizable topological space by a continuous one-to-one map. In this paper we answer two questions concerning sequence-covering maps on submetrizable spaces.

A test space is the set of outcome-sets associated with a collection of experiments. This notion provides a simple mathematical framework for the study of probabilistic theories -- notably, quantum mechanics -- in which one is faced with…

For every finite closure space $X$ one can define a finite topological space $\operatorname{Top} X$ together with a natural projection $\operatorname{Top} X\longrightarrow X$. This could allow to apply the techniques of topological…

The countable uniform power (or uniform box product) of a uniform space $X$ is a special topology on ${}^{\omega}X$ that lies between the Tychonoff topology and the box topology. We solve an open problem posed by P. Nyikos showing that if…

For a Tychonoff space $X$ and a family $\lambda$ of subsets of $X$, we denote by $C_{\lambda}(X)$ the space of all real-valued continuous functions on $X$ with the set-open topology. In this paper, we study the Menger and projective Menger…

We construct a complete locally convex topological vector space $X$ of countable algebraic dimension and a continuous linear operator $T:X\to X$ such that $T$ has no non-trivial closed invariant subspaces.

A base $\mathcal{B}$ for a space $X$ is said to be sharp if, whenever $x\in X$ and $(B_n)_{n\in\omega}$ is a sequence of pairwise distinct elements of $\mathcal{B}$ each containing $x$, the collection $\{\bigcap_{j\le n}B_j:n\in\omega\}$ is…

A new method is given for computing generators of the homology groups with integer coefficients for any finite $T_0$-space. An important role in this method is played by irreducible cycles which are defined here and give rise to continuous…

Recently many papers on cone metric spaces have been appeared, and main topological properties of such spaces have been obtained. A cone metric space is Hausdorff, and first countable, so the topology of it coincides with a topology induced…

We study the topology of metric spaces which are definable in o-minimal expansions of ordered fields. We show that a definable metric space either contains an infinite definable discrete set or is definably homeomorphic to a definable set…

We say that a Tychonoff space $X$ is a $\kappa$-space if it is homeomorphic to a closed subspace of $C_p(Y)$ for some locally compact space $Y$. The class of $\kappa$-spaces is strictly between the class of Dieudonn\'{e} complete spaces and…

Some results in C_k-theory are obtained with the use of bornologies. We investigate under which conditions the space of the continuous real functions with the compact-open topology is a productively countably tight space, which yields some…

A topological space $X$ is said to be an Ascoli space if any compact subset $K$ of $C_k(X)$ is evenly continuous. This definition is motivated by the classical Ascoli theorem. We study the $k_R$-property and the Ascoli property of…

The two main results of this work are the following: if a space $X$ is such that player II has a winning strategy in the game $\gone(\Omega_x, \Omega_x)$ for every $x \in X$, then $X$ is productively countably tight. On the other hand, if a…