If $X$ is a Tychonoff space, $C\left(X\right)$ its ring of real-valued continuous functions. In this paper, we study non-essential ideals in $C\left(X\right)$. Let $a$ be a infinite cardinal, then $X$ is called $a$-Kasch (resp. $\overline{a}$-Kasch) space if given any ideal (resp. $z$-ideal) $I$ with $gen\left(I\right)<a$ then $I$ is a non-essential ideal. We show that $X$ is an ${\aleph }_0$-Kasch space if and only if $X$ is an almost $P$-space and $X$ is an ${\aleph }_1$-Kasch space if and only if $X$ is a pseudocompact and almost $P$-space. Let $C_F\left(X\right)$ denote the socle of $C\left(X\right)$. For a topological space $X$ with only...

Assuming Martin’s Axiom, we provide an example of two Fréchet-Urysohn $\langle {\alpha }_4\rangle$-spaces, whose product is a non-Fréchet-Urysohn $\langle {\alpha }_4\rangle$-space. This gives a consistent negative answer to a question raised by T. Nogura.

An extremally disconnected space is called an absolute retract in the class of all extremally disconnected spaces if it is a retract of any extremally disconnected compact space in which it can be embedded. The Gleason spaces over dyadic spaces have this property. The main result of this paper says that if a space X of π-weight ${\omega }_1$ is an absolute retract in the class of all extremally disconnected compact spaces and X is homogeneous with respect to π-weight (i.e. all non-empty open sets have the same...

Let $X$ be the Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff non-single point spaces $X_{\alpha }$. Let $p\in X^{*}$ be a point in the closure of some $G\subset X$ whose weak Lindelöf number is strictly less than the cofinality of $\tau$. Then we show that $\beta X\setminus \left\{p\right\}$ is not normal. Under some additional assumptions, $p$ is a butterfly-point in $\beta X$. In particular, this is true if either $X={\omega }^{\tau }$ or $X=R^{\tau }$ and $\tau$ is infinite and not countably cofinal.

A scadic space is a Hausdorff continuous image of a product of compact scattered spaces. We complete a theorem begun by G. Chertanov that will establish that for each scadic space X, χ(X) = w(X). A ξ-adic space is a Hausdorff continuous image of a product of compact ordinal spaces. We introduce an either-or chain condition called Property $R_{\lambda }^{\text{'}}$ which we show is satisfied by all ξ-adic spaces. Whereas Property $R_{\lambda }^{\text{'}}$ is productive, we show that a weaker (but more natural) Property $R_{\lambda }$ is not productive. Polyadic...

Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kⁿ) or equivalently the n-fold injective tensor product $\otimes {̂}_{\epsilon }^{n}C\left(K\right)$ or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c₀(ω₁) in $\otimes {̂}_{\epsilon }^{n}C\left(K\right)$ under the hypothesis that C(K) contains such a copy. This is related to the results of E. Saab and P. Saab that $X\otimes {̂}_{\epsilon }Y$ contains a complemented copy of c₀ if one of the infinite-dimensional Banach...

We present an example of a connected, Polish, countable dense homogeneous space X that is not strongly locally homogeneous. In fact, a nontrivial homeomorphism of X is the identity on no nonempty open subset of X.