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It is shown that associated with each metric space (X,d) there is a compactification $u_{d}X$ of X that can be characterized as the smallest compactification of X to which each bounded uniformly continuous real-valued continuous function with domain X can be extended. Other characterizations of $u_{d}X$ are presented, and a detailed study of the structure of $u_{d}X$ is undertaken. This culminates in a topological characterization of the outgrowth $u_d{ℝ}^n\setminus {ℝ}^n$, where $({ℝ}^n,d)$ is Euclidean n-space with its usual metric.

The set of isolated points (resp. $P$-points) of a Tychonoff space $X$ is denoted by $Is\left(X\right)$ (resp. $P\left(X\right))$. Recall that $X$ is said to be if $Is\left(A\right)\ne ⌀$ whenever $⌀\ne A\subset X$. If instead we require only that $P\left(A\right)$ has nonempty interior whenever $⌀\ne A\subset X$, we say that $X$ is . Many theorems about scattered spaces hold or have analogs for spaces. For example, the union of a locally finite collection of SP-scattered spaces is SP-scattered. Some known theorems about Lindelöf or paracompact scattered spaces hold also in case the spaces are SP-scattered....

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...

If a metrizable space $X$ is dense in a metrizable space $Y$, then $Y$ is called a of $X$. If $T_1$ and $T_2$ are metric extensions of $X$ and there is a continuous map of $T_2$ into $T_1$ keeping $X$ pointwise fixed, we write $T_1\le T_2$. If $X$ is noncompact and metrizable, then $\left(ℳ\right(X),\le )$ denotes the set of metric extensions of $X$, where $T_1$ and $T_2$ are identified if $T_1\le T_2$ and $T_2\le T_1$, i.e., if there is a homeomorphism of $T_1$ onto $T_2$ keeping $X$ pointwise fixed. $\left(ℳ\right(X),\le )$ is a large complicated poset studied extensively by V. Bel’nov [, Trans. Moscow Math. Soc. (1975),...

Maximal pseudocompact spaces (i.e. pseudocompact spaces possessing no strictly stronger pseudocompact topology) are characterized. It is shown that submaximal pseudocompact spaces whose pseudocompact subspaces are closed need not be maximal pseudocompact. Various techniques for constructing maximal pseudocompact spaces are described. Maximal pseudocompactness is compared to maximal feeble compactness.