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3 results

PROOFS OF URYSOHN’S LEMMA AND THE TIETZE EXTENSION THEOREM VIA THE CANTOR FUNCTION
Part of
- Maps and general types of spaces defined by maps
- Fairly general properties
FLORICA C. CÎRSTEA
Journal:

Bulletin of the Australian Mathematical Society / Volume 103 / Issue 2 / April 2021

Published online by Cambridge University Press:

03 July 2020, pp. 326-332

Print publication:

April 2021
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Urysohn’s lemma is a crucial property of normal spaces that deals with separation of closed sets by continuous functions. It is also a fundamental ingredient in proving the Tietze extension theorem, another property of normal spaces that deals with the existence of extensions of continuous functions. Using the Cantor function, we give alternative proofs for Urysohn’s lemma and the Tietze extension theorem.

Approximation and extension of continuous functions
Part of
Salvador Hernández-Muñoz
Journal:

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics / Volume 57 / Issue 2 / October 1994

Published online by Cambridge University Press:

09 April 2009, pp. 149-157

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October 1994
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In this paper we study the approximation of vector valued continuous functions defined on a topological space and we apply this study to different problems. Thus we give a new proof of Machado's Theorem. Also we get a short proof of a Theorem of Katětov and we prove a generalization of Tietze's Extension Theorem for vector-valued continuous functions, thereby solving a question left open by Blair.

Natural Covers and R-Quotient Maps
S. M. Karnik, S. Willard
Journal:

Canadian Mathematical Bulletin / Volume 25 / Issue 4 / 01 December 1982

Published online by Cambridge University Press:

20 November 2018, pp. 456-461

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01 December 1982
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We extend the comprehensive treatment of k-spaces and sequential spaces provided by Franklin's refined notion of a natural cover to kR-spaces and sR-spaces. For this purpose, an apparently unstudied class of maps of topological spaces, the class of R-quotient maps, is introduced.

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3 results

PROOFS OF URYSOHN’S LEMMA AND THE TIETZE EXTENSION THEOREM VIA THE CANTOR FUNCTION

Approximation and extension of continuous functions

Natural Covers and R-Quotient Maps