On Discrete and Almost Discrete Topological Spaces

A topological space X is called almost discrete if every open subset of X is closed; equivalently, if every closed subset of X is open (comp. [6],[5]). Almost discrete spaces were investigated in Mizar formalism in [2]. We present here a few properties of such spaces supplementary to those given in [2]. Most interesting is the following characterization : A topological space X is almost discrete iff every nonempty subset of X is not nowhere dense. Hence, X is non almost discrete iff there is an everywhere dense subset of X different from the carrier of X. We have an analogous characterization of discrete spaces : A topological space X is discrete iff every nonempty subset of X is not boundary. Hence, X is non discrete iff there is a dense subset of X different from the carrier of X. It is well known that the class of all almost discrete spaces contains both the class of discrete spaces and the class of anti-discrete spaces (see e.g., [2]). Observations presented here show that the class of all almost discrete non discrete spaces is not contained in the class of anti-discrete spaces and the class of all almost discrete non anti-discrete spaces is not contained in the class of discrete spaces. Moreover, the class of almost discrete non discrete non anti-discrete spaces is nonempty. To analyse these interdependencies we use various examples of topological spaces constructed here in Mizar formalism.