Locally finite families, completely separated sets and remote points

Abstract:

It is shown that if $X$ is a nonpseudocompact space with a $\sigma$-locally finite $\pi$-base, then $X$ has remote points. Within the class of spaces possessing a $\sigma$-locally finite $\pi$-base, this result extends the work of Chae and Smith, because their work utilized normality to achieve complete separation. It provides spaces which have remote points, where the spaces do not satisfy the conditions required in the previous works by Dow, by van Douwen, by van Mill, or by Peters. The lemma: "Let $X$ be a space and let $\{ {C_\xi }:\xi < \alpha \}$ be a locally finite family of cozero sets of $X$. Let $\{ {Z_\xi }:\xi < \alpha \}$ be a family of zero sets of $X$ such that for each $\xi < \alpha ,{Z_\xi } \subset {C_\xi }$. Then ${ \cup _{\xi < \alpha }}{Z_\xi }$ is completely separated from $X/{ \cup _{\xi < \alpha }}{C_\xi }$", is a fundamental tool in this work. An example is given which demonstrates the value of this tool. The example also refutes an appealing conjecture—a conjecture for which the authors found that there existed significant confusion within the topological community as to its truth or falsity.