Journal of Formalized Mathematics
Volume 7, 1995
University of Bialystok
Copyright (c) 1995 Association of Mizar Users

## Dyadic Numbers and T\$_4\$ Topological Spaces

Jozef Bialas
Lodz \ University, Lodz
Yatsuka Nakamura
Shinshu University, Nagano

### Summary.

This article is the first part of a paper proving the fundamental Urysohn's Theorem concerning the existence of a real valued continuous function on a normal topological space. The paper is divided into four parts. In the first part, we prove some auxiliary theorems concerning properties of natural numbers and prove two useful schemes about recurrently defined functions; in the second part, we define a special set of rational numbers, which we call dyadic, and prove some of its properties. The next part of the paper contains the definitions of T\${}_1\$ space and normal space, and we prove related theorems used in later parts of the paper. The final part of this work is developed for proving the theorem about the existence of some special family of subsets of a topological space. This theorem is essential in proving Urysohn's Lemma.

#### MML Identifier: URYSOHN1

The terminology and notation used in this paper have been introduced in the following articles [13] [6] [15] [14] [12] [4] [5] [9] [3] [1] [8] [7] [11] [2] [10]

#### Contents (PDF format)

2. Normal Spaces
3. Some Increasing Family of Sets in Normal Space

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