Continuous Selections, Function Spaces and Partitions of Unity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908851862765568 |
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| author | Gutev, Valentin |
| author_facet | Gutev, Valentin |
| contents | The famous Michael selection theorem deals with the characterisation of paracompact spaces by continuous selections of lower semi-continuous mappings in Banach spaces. In this paper, we will discuss several equivalent forms of this theorem, without explicitly mentioning paracompactness. This will be based on a previous result, also obtained by Michael, that a space $X$ is paracompact if and only if every open cover of $X$ has an index-subordinated partition of unity. Thus, we will show that the existence of such partitions of unity on a space $X$ is equivalent to the existence of continuous selections for special lower semi-continuous mappings from $X$ to the nonempty convex subsets of special function spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21313 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Continuous Selections, Function Spaces and Partitions of Unity Gutev, Valentin Functional Analysis 54B10, 54C30, 54C35, 54C60, 54C65, 55U10 The famous Michael selection theorem deals with the characterisation of paracompact spaces by continuous selections of lower semi-continuous mappings in Banach spaces. In this paper, we will discuss several equivalent forms of this theorem, without explicitly mentioning paracompactness. This will be based on a previous result, also obtained by Michael, that a space $X$ is paracompact if and only if every open cover of $X$ has an index-subordinated partition of unity. Thus, we will show that the existence of such partitions of unity on a space $X$ is equivalent to the existence of continuous selections for special lower semi-continuous mappings from $X$ to the nonempty convex subsets of special function spaces. |
| title | Continuous Selections, Function Spaces and Partitions of Unity |
| topic | Functional Analysis 54B10, 54C30, 54C35, 54C60, 54C65, 55U10 |
| url | https://arxiv.org/abs/2602.21313 |