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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.14755 |
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| _version_ | 1866909263150972928 |
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| author | Nxumalo, Mbekezeli |
| author_facet | Nxumalo, Mbekezeli |
| contents | In locale theory, a sublocale is said to be remote in case it misses every nowhere dense sublocale. In this paper, we introduce and study a new class of sublocales in the category of bilocales, namely (i,j)-remote sublocales. These are bilocalic counterparts of remote sublocales and are the sublocales missing every (i,j)-nowhere dense sublocale, with (i,j)-nowhere dense sublocales being bilocalic counterparts of (τ_{i},τ_{j})-nowhere dense subsets in bitopological spaces. A comprehensive study of (i,j)-nowhere dense sublocales is given and we show that in the class of balanced bilocales, a sublocale is (i,j)-nowhere dense if and only if its bilocale closure is nowhere dense. We also consider weakly (i,j)-remote sublocales which are those sublocales missing every clopen (i,j)-nowhere dense sublocale. Furthermore, we extend (i,j)-remoteness to the categories of bitopological spaces as well as normed lattices. In the class of \sup-T_{D} bitopological spaces, a subset A of a bitopological space (X,τ_{1},τ_{2}) is (τ_{i},τ_{j})-remote if and only if the induced sublocale \widetilde{A} of τ_{1}\veeτ_{2} is (i,j)-remote. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14755 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remoteness in the category of bilocales Nxumalo, Mbekezeli General Topology 06D22 In locale theory, a sublocale is said to be remote in case it misses every nowhere dense sublocale. In this paper, we introduce and study a new class of sublocales in the category of bilocales, namely (i,j)-remote sublocales. These are bilocalic counterparts of remote sublocales and are the sublocales missing every (i,j)-nowhere dense sublocale, with (i,j)-nowhere dense sublocales being bilocalic counterparts of (τ_{i},τ_{j})-nowhere dense subsets in bitopological spaces. A comprehensive study of (i,j)-nowhere dense sublocales is given and we show that in the class of balanced bilocales, a sublocale is (i,j)-nowhere dense if and only if its bilocale closure is nowhere dense. We also consider weakly (i,j)-remote sublocales which are those sublocales missing every clopen (i,j)-nowhere dense sublocale. Furthermore, we extend (i,j)-remoteness to the categories of bitopological spaces as well as normed lattices. In the class of \sup-T_{D} bitopological spaces, a subset A of a bitopological space (X,τ_{1},τ_{2}) is (τ_{i},τ_{j})-remote if and only if the induced sublocale \widetilde{A} of τ_{1}\veeτ_{2} is (i,j)-remote. |
| title | Remoteness in the category of bilocales |
| topic | General Topology 06D22 |
| url | https://arxiv.org/abs/2407.14755 |