We show that if $X$ is first-countable, of countable extent, and a subspace of some ordinal, then $C_p(X,2)$ is Lindelöf.

We investigate spaces $C_p(·,n)$ over LOTS (linearly ordered topological spaces). We find natural necessary conditions for linear Lindelöfness of $C_p(·,n)$ over LOTS. We also characterize countably compact LOTS whose $C_p(·,n)$ is linearly Lindelöf for each n. Both the necessary conditions and the characterization are given in terms of the topology of the Dedekind completion of a LOTS.

In this paper, we prove the following statements: (1) For every regular uncountable cardinal $\kappa$, there exist a Tychonoff space $X$ and $Y$ a subspace of $X$ such that $Y$ is both relatively absolute star-Lindelöf and relative property (a) in $X$ and $e(Y,X)\ge \kappa$, but $Y$ is not strongly relative star-Lindelöf in $X$ and $X$ is not star-Lindelöf. (2) There exist a Tychonoff space $X$ and a subspace $Y$ of $X$ such that $Y$ is strongly relative star-Lindelöf in $X$ (hence, relative star-Lindelöf), but $Y$ is not absolutely relative star-Lindelöf...

In this paper, we prove the following statements: (1) For any cardinal $\kappa$, there exists a Tychonoff star-Lindelöf space $X$ such that $a\left(X\right)\ge \kappa$. (2) There is a Tychonoff discretely star-Lindelöf space $X$ such that $aa\left(X\right)$ does not exist. (3) For any cardinal $\kappa$, there exists a Tychonoff pseudocompact $\sigma$-starcompact space $X$ such that $st\text{-}l\left(X\right)\ge \kappa$.

We prove that if $X$ is a first countable space with property $\left(DC\left({\omega }_1\right)\right)$ and with a $G_{\delta }$-diagonal then the cardinality of $X$ is at most $𝔠$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$.

For a topological property $P$, we say that a space $X$ is star $P$ if for every open cover $𝒰$ of the space $X$ there exists $A\subset X$ such that $st(A,𝒰)=X$. We consider space with star countable extent establishing the relations between the star countable extent property and the properties star Lindelöf and feebly Lindelöf. We describe some classes of spaces in which the star countable extent property is equivalent to either the Lindelöf property or separability. An example is given of a Tychonoff star Lindelöf space with...

The authors give a ZFC example for a space with $\mathit{\text{Split}}(\Omega ,\Omega )$ but not $\mathit{\text{Split}}(\Lambda ,\Lambda )$.

If Nonempty has a winning strategy against Empty in the Choquet game on a space, the space is said to be a Choquet space. Such a winning strategy allows Nonempty to consider the entire finite history of previous moves before making each new move; a stationary strategy only permits Nonempty to consider the previous move by Empty. We show that Nonempty has a stationary winning strategy for every second-countable T₁ Choquet space. More generally, Nonempty has a stationary winning strategy for...

A space $X$ is said to be strongly base-paracompact if there is a basis $ℬ$ for $X$ with $\left|ℬ\right|=w\left(X\right)$ such that every open cover of $X$ has a star-finite open refinement by members of $ℬ$. Strongly paracompact spaces which are strongly base-paracompact are studied. Strongly base-paracompact spaces are shown have a family of functions $ℱ$ with cardinality equal to the weight such that every open cover has a locally finite partition of unity subordinated to it from $ℱ$.

Strongly paracompact metrizable spaces are characterized in terms of special S-maps onto metrizable non-Archimedean spaces. A similar characterization of strongly metrizable spaces is obtained as well. The approach is based on a sieve-construction of "metric"-continuous pseudo-sections of lower semicontinuous mappings.