An observation on spaces with a zeroset diagonal
We say that a space $X$ has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. A space $X$ has a zeroset diagonal if there is a continuous mapping $f X^2 \rightarrow[0,1]$ with $\Delta_X=f^{-1}(0)$, where $\Delta_X=\{(x,x)\c...
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2020-04-01
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| Series: | Mathematica Bohemica |
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| Online Access: | http://mb.math.cas.cz/full/145/1/mb145_1_2.pdf |
| _version_ | 1828823034789625856 |
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| author | Wei-Feng Xuan |
| author_facet | Wei-Feng Xuan |
| author_sort | Wei-Feng Xuan |
| collection | DOAJ |
| description | We say that a space $X$ has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. A space $X$ has a zeroset diagonal if there is a continuous mapping $f X^2 \rightarrow[0,1]$ with $\Delta_X=f^{-1}(0)$, where $\Delta_X=\{(x,x)\colon x\in X\}$. In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most $\mathfrak c$. |
| first_indexed | 2024-12-12T13:26:50Z |
| format | Article |
| id | doaj.art-02f42e41dd124910a8dc5844cae0dd7b |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-12-12T13:26:50Z |
| publishDate | 2020-04-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-02f42e41dd124910a8dc5844cae0dd7b2022-12-22T00:23:10ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362020-04-011451151810.21136/MB.2018.0016-18MB.2018.0016-18An observation on spaces with a zeroset diagonalWei-Feng XuanWe say that a space $X$ has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. A space $X$ has a zeroset diagonal if there is a continuous mapping $f X^2 \rightarrow[0,1]$ with $\Delta_X=f^{-1}(0)$, where $\Delta_X=\{(x,x)\colon x\in X\}$. In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most $\mathfrak c$.http://mb.math.cas.cz/full/145/1/mb145_1_2.pdf first countable discrete countable chain condition zeroset diagonal cardinal |
| spellingShingle | Wei-Feng Xuan An observation on spaces with a zeroset diagonal Mathematica Bohemica first countable discrete countable chain condition zeroset diagonal cardinal |
| title | An observation on spaces with a zeroset diagonal |
| title_full | An observation on spaces with a zeroset diagonal |
| title_fullStr | An observation on spaces with a zeroset diagonal |
| title_full_unstemmed | An observation on spaces with a zeroset diagonal |
| title_short | An observation on spaces with a zeroset diagonal |
| title_sort | observation on spaces with a zeroset diagonal |
| topic | first countable discrete countable chain condition zeroset diagonal cardinal |
| url | http://mb.math.cas.cz/full/145/1/mb145_1_2.pdf |
| work_keys_str_mv | AT weifengxuan anobservationonspaceswithazerosetdiagonal AT weifengxuan observationonspaceswithazerosetdiagonal |