Step functions from the half-open unit interval into a topological space |
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Authors: | Kevin Bicknell |
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Institution: | Department of Mathematics, La Trobe University, Bundoora 3083, Australia |
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Abstract: | The set of continuous-from-the-right step functions from the half-open unit interval0, 1into a topological space X is denoted by X1. Elsewhere a topology has been defined which makes X1 a contractible, locally contractible space with the subspace of constant functions being homeomorphic to X. When X has a bounded metric ?, the topology of X1 may be described by the metric .It is shown here that if X is separable, then X1 is separable and if X satisfies the first (or second) axiom of countability, then X1 satisfies it too. In contrast, it is shown that properties such as normality do not extend from X to X1. This follows from the main result: X1 is homeomorphic to its square, and thus contains a copy of X×X (which is closed when X is Hausdorff). The final theorem states that if X has at least two points then X1 is not complete metrizable. |
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Keywords: | 54C25 54C60 |
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