Volume 4, Issue 2, April 2016, Page: 75-79
Hypographs of Upper Semi-continuous Maps and Continuous Maps on a Bounded Open Interval
Nada Wu, Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, Guangdong, China
Received: Mar. 23, 2016;       Published: Mar. 25, 2016
DOI: 10.11648/j.ajam.20160402.12      View  3223      Downloads  89
Abstract
For any bounded open interval X in the Euclidean space E1, let ↓USC(X) and ↓C(X) be the families of all hypographs of upper semi-continuous maps and continuous maps from X to I=[0,1], respectively. They are endowed with the topology induced by the Hausdorff metric of the metric space Y×I,Y is the closure of X. It was proved in other two papers respectively that ↓USC(X) and ↓C(X) are homeomorphic to s and c0 respectively, where s=(-1,1) and c0={(xn)ϵ(-1,1) : limn→ ∞ xn=0}. However the topological structure of the pair (↓USC(X), ↓C(X)) was not clear. In the present paper, it is proved in the strongly universal method that the pair of spaces (↓USC(X), ↓C(X)) is pair homeomorphic to (s,c0) which is not homeomorphic to (s, c0). Hence this paper figures out the topological structure of the pair (↓USC(X), ↓C(X)).
Keywords
Hypograph, Upper Semi-continuous Maps, Continuous Maps, Bounded Open Interval, Hausdorff Metric, The Property of Strongly Universal
To cite this article
Nada Wu, Hypographs of Upper Semi-continuous Maps and Continuous Maps on a Bounded Open Interval, American Journal of Applied Mathematics. Vol. 4, No. 2, 2016, pp. 75-79. doi: 10.11648/j.ajam.20160402.12
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