Journal of Formalized Mathematics
Volume 5, 1993
University of Bialystok
Copyright (c) 1993 Association of Mizar Users

## On Discrete and Almost Discrete Topological Spaces

Zbigniew Karno
Warsaw University, Bialystok

### Summary.

A topological space $X$ is called {\em almost discrete}\/ if every open subset of $X$ is closed; equivalently, if every closed subset of $X$ is open (comp. [6],[7]). Almost discrete spaces were investigated in Mizar formalism in [4]. We present here a few properties of such spaces supplementary to those given in [4].\par Most interesting is the following characterization~: {\em A topological space $X$ is almost discrete iff every nonempty subset of $X$ is not nowhere dense}. Hence, {\em $X$ 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~: {\em A topological space $X$ is discrete iff every nonempty subset of $X$ is not boundary}. Hence, {\em $X$ 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., [4]). 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.

#### MML Identifier: TEX_1

The terminology and notation used in this paper have been introduced in the following articles [10] [11] [8] [1] [9] [12] [3] [4] [5] [2]

#### Contents (PDF format)

1. Properties of Subsets of a Topological Space with Modified Topology
2. Trivial Topological Spaces
3. Examples of Discrete and Anti-discrete Topological Spaces
4. An Example of a Topological Space
5. Discrete and Almost Discrete Spaces

#### Acknowledgments

The author wishes to thank to Professor A. Trybulec for many helpful conversations during the preparation of this paper.

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