Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

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

- 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.

- Real Numbers
- Topological Preliminaries
- Points and Subsets in ${\cal E}^2_{\rm T}$
- Balls as subsets of ${\cal E}^n_{\rm T}$
- Topological Properties of Real Numbers Subsets
- Bounded Subsets

- [1]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek.
K\"onig's theorem.
*Journal of Formalized Mathematics*, 2, 1990. - [4]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Leszek Borys.
Paracompact and metrizable spaces.
*Journal of Formalized Mathematics*, 3, 1991. - [6]
Czeslaw Bylinski.
Binary operations.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [11]
Czeslaw Bylinski.
The modification of a function by a function and the iteration of the composition of a function.
*Journal of Formalized Mathematics*, 2, 1990. - [12]
Czeslaw Bylinski.
The sum and product of finite sequences of real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [13]
Czeslaw Bylinski and Piotr Rudnicki.
Bounding boxes for compact sets in $\calE^2$.
*Journal of Formalized Mathematics*, 9, 1997. - [14]
Agata Darmochwal.
Compact spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Agata Darmochwal.
Families of subsets, subspaces and mappings in topological spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [16]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Agata Darmochwal.
The Euclidean space.
*Journal of Formalized Mathematics*, 3, 1991. - [18]
Agata Darmochwal and Yatsuka Nakamura.
Metric spaces as topological spaces --- fundamental concepts.
*Journal of Formalized Mathematics*, 3, 1991. - [19]
Agata Darmochwal and Yatsuka Nakamura.
The topological space $\calE^2_\rmT$. Arcs, line segments and special polygonal arcs.
*Journal of Formalized Mathematics*, 3, 1991. - [20]
Alicia de la Cruz.
Totally bounded metric spaces.
*Journal of Formalized Mathematics*, 3, 1991. - [21]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [22]
Stanislawa Kanas, Adam Lecko, and Mariusz Startek.
Metric spaces.
*Journal of Formalized Mathematics*, 2, 1990. - [23]
Jaroslaw Kotowicz.
Convergent real sequences. Upper and lower bound of sets of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [24]
Jaroslaw Kotowicz.
Convergent sequences and the limit of sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [25]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
*Journal of Formalized Mathematics*, 1, 1989. - [26]
Rafal Kwiatek.
Factorial and Newton coefficients.
*Journal of Formalized Mathematics*, 2, 1990. - [27]
Yatsuka Nakamura and Czeslaw Bylinski.
Extremal properties of vertices on special polygons, part I.
*Journal of Formalized Mathematics*, 6, 1994. - [28]
Yatsuka Nakamura and Jaroslaw Kotowicz.
The Jordan's property for certain subsets of the plane.
*Journal of Formalized Mathematics*, 4, 1992. - [29]
Yatsuka Nakamura, Andrzej Trybulec, and Czeslaw Bylinski.
Bounded domains and unbounded domains.
*Journal of Formalized Mathematics*, 11, 1999. - [30]
Takaya Nishiyama and Yasuho Mizuhara.
Binary arithmetics.
*Journal of Formalized Mathematics*, 5, 1993. - [31]
Beata Padlewska.
Connected spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [32]
Beata Padlewska.
Locally connected spaces.
*Journal of Formalized Mathematics*, 2, 1990. - [33]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [34]
Jan Popiolek.
Some properties of functions modul and signum.
*Journal of Formalized Mathematics*, 1, 1989. - [35]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [36]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [37]
Andrzej Trybulec.
A Borsuk theorem on homotopy types.
*Journal of Formalized Mathematics*, 3, 1991. - [38]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [39]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
*Journal of Formalized Mathematics*, 1, 1989. - [40]
Andrzej Trybulec and Yatsuka Nakamura.
On the rectangular finite sequences of the points of the plane.
*Journal of Formalized Mathematics*, 9, 1997. - [41]
Wojciech A. Trybulec.
Pigeon hole principle.
*Journal of Formalized Mathematics*, 2, 1990. - [42]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [43]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [44]
Mariusz Zynel and Adam Guzowski.
\Tzero\ topological spaces.
*Journal of Formalized Mathematics*, 6, 1994.

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