Journal of Formalized Mathematics
Volume 11, 1999
University of Bialystok
Copyright (c) 1999 Association of Mizar Users

## Compactness of the Bounded Closed Subsets of $\cal E^2_\rm T$

Artur Kornilowicz
University of Bialystok
This paper was written while the author visited Shinshu University, winter 1999.

### Summary.

This paper contains theorems which describe the correspondence between topological properties of real numbers subsets introduced in [35] and introduced in [33], [14]. We also show the homeomorphism between the cartesian product of two $R^1$ and ${\cal E}^2_{\rm T}$. The compactness of the bounded closed subset of ${\cal E}^2_{\rm T}$ is proven.

#### MML Identifier: TOPREAL6

The terminology and notation used in this paper have been introduced in the following articles [36] [9] [42] [43] [7] [8] [6] [16] [2] [38] [21] [39] [1] [34] [30] [11] [25] [24] [23] [22] [20] [4] [10] [12] [26] [3] [41] [35] [33] [15] [31] [32] [14] [37] [5] [17] [18] [19] [44] [28] [13] [27] [40] [29]

#### Contents (PDF format)

1. Real Numbers
2. Topological Preliminaries
3. Points and Subsets in ${\cal E}^2_{\rm T}$
4. Balls as subsets of ${\cal E}^n_{\rm T}$
5. Topological Properties of Real Numbers Subsets
6. Bounded Subsets

#### Acknowledgments

I would like to thank Professor Yatsuka Nakamura for his help in the preparation of the article.

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