Volume 9, 1997

University of Bialystok

Copyright (c) 1997 Association of Mizar Users

**Roman Matuszewski**- University of Bialystok
- The work was done, while the author stayed at Nagano in the fall of 1996.
**Yatsuka Nakamura**- Shinshu University, Nagano

- In the $n$-dimensional Euclidean space ${\cal E}^n_{\rm T}$, a projection operator to each coordinate is defined. It is proven that such an operator is linear. Moreover, it is continuous as a mapping from ${\cal E}^n_{\rm T}$ to ${R}^{1}$, the carrier of which is a set of all reals. If $n$ is 1, the projection becomes a homeomorphism, which means that ${\cal E}^1_{\rm T}$ is homeomorphic to ${R}^{1}$.

- Projections
- Continuity of Projections
- 1-dimensional and 2-dimensional Cases

- [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 and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Grzegorz Bancerek and Andrzej Trybulec.
Miscellaneous facts about functions.
*Journal of Formalized Mathematics*, 8, 1996. - [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.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [10]
Czeslaw Bylinski.
The sum and product of finite sequences of real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [11]
Czeslaw Bylinski and Piotr Rudnicki.
Bounding boxes for compact sets in $\calE^2$.
*Journal of Formalized Mathematics*, 9, 1997. - [12]
Agata Darmochwal.
Families of subsets, subspaces and mappings in topological spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Agata Darmochwal.
The Euclidean space.
*Journal of Formalized Mathematics*, 3, 1991. - [14]
Agata Darmochwal and Yatsuka Nakamura.
Metric spaces as topological spaces --- fundamental concepts.
*Journal of Formalized Mathematics*, 3, 1991. - [15]
Agata Darmochwal and Yatsuka Nakamura.
The topological space $\calE^2_\rmT$. Arcs, line segments and special polygonal arcs.
*Journal of Formalized Mathematics*, 3, 1991. - [16]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Stanislawa Kanas, Adam Lecko, and Mariusz Startek.
Metric spaces.
*Journal of Formalized Mathematics*, 2, 1990. - [18]
Jaroslaw Kotowicz.
Functions and finite sequences of real numbers.
*Journal of Formalized Mathematics*, 5, 1993. - [19]
Takaya Nishiyama and Yasuho Mizuhara.
Binary arithmetics.
*Journal of Formalized Mathematics*, 5, 1993. - [20]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [21]
Jan Popiolek.
Some properties of functions modul and signum.
*Journal of Formalized Mathematics*, 1, 1989. - [22]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [23]
Andrzej Trybulec.
A Borsuk theorem on homotopy types.
*Journal of Formalized Mathematics*, 3, 1991. - [24]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [25]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
*Journal of Formalized Mathematics*, 1, 1989. - [26]
Wojciech A. Trybulec.
Pigeon hole principle.
*Journal of Formalized Mathematics*, 2, 1990. - [27]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [28]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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