Countable Connected Spaces

Abstract

Two pathological countable topological spaces are constructed. Each is quasimetrizable and has a simple explicit quasimetric. One is a locally connected Hausdorff space and is an extension of the rationals. The other is a connected space which becomes totally disconnected upon the removal of a single point. This space satisfies the Urysohn separation property-a property between Tí and T¡-and is an extension of the space of rational points in the plane. Both are one dimensional in the Menger-Urysohn [inductive] sense and infinite dimensional in the Lebesgue [covering] sense.

The Definition and Properties of New Connectedness in Topology Space

“…The answer is affirmative. Roy, Kallnan [6], Miller [7] all made related constructions. Bing [8] constructed a countable Hausdorff connected space but it was not strongly connected space.…”

Continuous Functions from a Connected Locally Connected Space into a Connected Space with a Dispersion Point

“…There are many examples of connected spaces with a dispersion point, a classic one given by Knaster and Kuratowski [1] and for others, see [2], [3], [4]. For terminology, see [6].…”