In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of εn with a non-empty interior. This theorem states that, if T is a normal topological space, X is a closed subset of T, and A is a convex compact subset of εn with a non-empty interior, then a continuous function f : X → A can be extended to a continuous function g : T → εn. Additionally we show that a subset A is replaceable by an arbitrary subset of a topological space that is homeomorphic with a convex compact subset of En with a non-empty interior. This article is based on [20]; [23] and [22] can also serve as reference books.
@article{bwmeta1.element.doi-10_2478_forma-2014-0002,
author = {Karol P\k ak},
title = {Tietze Extension Theorem for n-dimensional Spaces},
journal = {Formalized Mathematics},
volume = {22},
year = {2014},
pages = {11-19},
zbl = {1298.54003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_forma-2014-0002}
}
Karol Pąk. Tietze Extension Theorem for n-dimensional Spaces. Formalized Mathematics, Tome 22 (2014) pp. 11-19. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_forma-2014-0002/
[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
[2] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589-593, 1990.
[3] Grzegorz Bancerek. Cartesian product of functions. Formalized Mathematics, 2(4):547-552, 1991.
[4] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. | Zbl 06213858
[5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
[6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
[7] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.
[8] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
[9] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.
[10] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
[11] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
[12] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
[13] Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.
[14] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
[15] Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383-386, 1990.
[16] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.
[17] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
[18] Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.
[19] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.
[20] Roman Duda. Wprowadzenie do topologii. PWN, 1986. | Zbl 0636.54001
[21] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53-58, 2003.
[22] Ryszard Engelking. Dimension Theory. North-Holland, Amsterdam, 1978.
[23] Ryszard Engelking. General Topology. Heldermann Verlag, Berlin, 1989.
[24] Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4): 449-454, 1997.
[25] Artur Korniłowicz. The correspondence between n-dimensional Euclidean space and the product of n real lines. Formalized Mathematics, 18(1):81-85, 2010. doi:10.2478/v10037-010-0011-0.[Crossref]
[26] Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181-187, 2005.
[27] Artur Korniłowicz. Homeomorphism between [:EiT , EjT :] and Ei+jT . Formalized Mathematics, 8(1):73-76, 1999.
[28] Artur Korniłowicz. On the continuity of some functions. Formalized Mathematics, 18(3): 175-183, 2010. doi:10.2478/v10037-010-0020-z.[Crossref]
[29] Artur Korniłowicz. Arithmetic operations on functions from sets into functional sets. Formalized Mathematics, 17(1):43-60, 2009. doi:10.2478/v10037-009-0005-y.[Crossref]
[30] Artur Korniłowicz and Yasunari Shidama. Brouwer fixed point theorem for disks on the plane. Formalized Mathematics, 13(2):333-336, 2005.
[31] Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in En T. Formalized Mathematics, 12(3):301-306, 2004.
[32] Yatsuka Nakamura, Andrzej Trybulec, and Czesław Bylinski. Bounded domains and unbounded domains. Formalized Mathematics, 8(1):1-13, 1999.
[33] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
[34] Karol Pak. Basic properties of metrizable topological spaces. Formalized Mathematics, 17(3):201-205, 2009. doi:10.2478/v10037-009-0024-8.[Crossref]
[35] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.
[36] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4): 535-545, 1991.
[37] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.
[38] Andrzej Trybulec. On the geometry of a Go-Board. Formalized Mathematics, 5(3):347-352, 1996.
[39] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.
[40] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
[41] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
[42] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
[43] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990.
[44] Mariusz Zynel and Adam Guzowski. T0 topological spaces. Formalized Mathematics, 5 (1):75-77, 1996.