convex compact subset
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2014 ◽  
Vol 22 (1) ◽  
pp. 11-19 ◽  
Author(s):  
Karol Pąk

Summary 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.



2011 ◽  
Vol 19 (3) ◽  
pp. 151-153 ◽  
Author(s):  
Karol Pąk

Brouwer Fixed Point Theorem in the General Case In this article we prove the Brouwer fixed point theorem for an arbitrary convex compact subset of εn with a non empty interior. This article is based on [15].



1988 ◽  
Vol 37 (2) ◽  
pp. 221-225 ◽  
Author(s):  
A. Sersouri

We prove that if X and Y are two (real) Banach spaces such that dim X ≥ 2 and dim Y ≥ 2, then the space K(X, Y) contains a convex compact subset C with dim C ≥ 2 (in the affine sense) which fails to be an intersection of balls. This improves two results of Ruess and Stegall.



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