## Doctoral Dissertations

Winter 1990

Dissertation

Mathematics

#### Degree Name

Doctor of Philosophy

Samuel D Shore

#### Abstract

This work investigates the use of distance function constructions in the study of semimetrizable spaces, especially as this relates to developable, K-semimetrizable and 1-continuously semimetrizable spaces.

A distance function for X is a nonnegative, symmetric, real-valued function d: X x X $\to$ $\IR$ such that d(p,q) = 0 iff p = q. A distance function d is developable iff, when d(x$\sb{\rm n}$,p) $\to$ 0 and d(y$\sb{\rm n}$,p) $\to$ 0, then d(x$\sb{\rm n}$,y$\sb{\rm n}$) $\to$ 0; and d is a K-distance function iff whenever d(x$\sb{\rm n}$,p) $\to$ 0, d(y$\sb{\rm n}$,q) $\to$ 0 and d(x$\sb{\rm n}$,y$\sb{\rm n}$) $\to$ 0, then p = q.

A topological space (X,${\cal J}$) is semimetrizable iff there is a distance function d for X such that, for every A $\subseteq$ X, d-cl(A) = A$\sp{\cal J}$. A topological space is developable (resp. K-, 1-continuously) semimetrizable when d is a developable (resp. K-, 1-continuous) distance function.

First, we use our approach to prove the classical metrization theorems. Then, in searching for new results, we establish characterizations involving sequences of open covers and diagonal conditions.

Theorem. (X,${\cal J}$) is Hausdorff and developable semimetrizable iff it is a w$\Delta$-space with a G$\sb\delta\sp\*$-diagonal.

Theorem. (X,${\cal J}$) is K-developable semimetrizable iff it is a w$\Delta$-space with a regular G$\sb\delta$-diagonal.

We conclude our study with characterizations which are given in terms of neighborhood structures; $\{$U$\sb{\rm n}$(p): n $\in$ $\rm I\!N$, p $\in$ X$\}$ is a neighborhood structure for (X,${\cal J}$) iff p $\in$ U$\sb{\rm n}$(p) $\in$ ${\cal J}$ and U$\sb{\rm n+1}$(p) $\subseteq$ U$\sb{\rm n}$(p), for every n $\in$ $\rm I\!N$. We characterize open, K- and developable semimetrizable spaces. For example,

Theorem. (X,${\cal J}$) is developable semimetrizable iff there is a neighborhood structure $\{$U$\sb{\rm n}$(p): n $\in$ $\rm I\!N$, p $\in$ X$\}$ for (X,${\cal J}$) such that: (i) $\cap\{$U$\sb{\rm n}$(p): n $\in$ $\rm I\!N\}$ = $\{$p$\}$; and (ii) if x$\sb{\rm n}$, p $\in$ U$\sb{\rm n}$(y$\sb{\rm n}$) for some y$\sb{\rm n}$ $\in$ X, then x$\sb{\rm n} \to$ p (in ${\cal J}$).

In retrospect, we have found new characterizations or improved old characterizations of developable semimetrizable spaces and other more restricted kinds of developable spaces, while our study of 1-continuously semimetrizable spaces remains quite incomplete.

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