A nonstandard approach to realcompactness with applications to the equation ν(X × Y) = ν(X) × ν(Y)
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- 作者：Render ; Hermann
- 关键词：Realcompactification ; Topological completion ; Biquotient map ; Bi-k-space ; 54J05 ; 54D60
- 刊名：Topology and its Applications
- 出版年：1996
- 出版时间：February 29, 1996
- 年：1996
- 卷：68
- 期：3
- 页码：205-239
- 全文大小：1.70 M
文摘
The set psns*X of all pseudonearstandard points and its relationship to topological concepts will be systemically investigated. For example we obtain nonstandard characterizations for the property that the realcompactification 
(X) is locally compact or a bi-k-space. The study of the equation psns* (X × Y) = psns* X × psns*Y is equivalent to the question whether the extension of the identity map id: X × Y → (X) × (Y) to the space (X × Y) is a biquotient map and sufficient conditions are proved. Moreover we show that this question is not equivalent to the validity of the equation (X × Y) = (X) × (Y) under the assumption of the existence of a measurable cardinal. If (X) and (Y) are bi-k-spaces both equations are equivalent to the property that the product of two bounded subsets is bounded.
(X) is locally compact or a bi-k-space. The study of the equation psns* (X × Y) = psns* X × psns*Y is equivalent to the question whether the extension of the identity map id: X × Y → (X) × (Y) to the space (X × Y) is a biquotient map and sufficient conditions are proved. Moreover we show that this question is not equivalent to the validity of the equation (X × Y) = (X) × (Y) under the assumption of the existence of a measurable cardinal. If (X) and (Y) are bi-k-spaces both equations are equivalent to the property that the product of two bounded subsets is bounded.