Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.12450 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915852917407744 |
|---|---|
| author | Kuba, Gerald |
| author_facet | Kuba, Gerald |
| contents | Let S denote the family of all subspaces of the plane that are graphs of functions from the real line R to itself. We prove that S has two subfamilies G,H of spaces such that the cardinality of G is c (the cardinality of the continuum) and the cardinality of H is 2^c, every space in the family G is completely metrizable, each element of H is a dense subset of the plane and the elements of the union of G and H are pairwise non-embeddable (i.p. pairwise non-homeomorphic) subspaces of the plane. On the other hand, the family S contains precisely countably infinitely many locally connected spaces up to homeomorphism, and if X,Y are such spaces then X is embeddable into Y. Furthermore, if T is a topology on the set R finer than the Euclidean topology and the space (R,T) is separable and locally connected then the space is locally compact and homeomorphic to some space in S. In a very natural way we establish a complete classification of all these refinements T of the real line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12450 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On real functions with graphs either connected or locally connected Kuba, Gerald General Topology Let S denote the family of all subspaces of the plane that are graphs of functions from the real line R to itself. We prove that S has two subfamilies G,H of spaces such that the cardinality of G is c (the cardinality of the continuum) and the cardinality of H is 2^c, every space in the family G is completely metrizable, each element of H is a dense subset of the plane and the elements of the union of G and H are pairwise non-embeddable (i.p. pairwise non-homeomorphic) subspaces of the plane. On the other hand, the family S contains precisely countably infinitely many locally connected spaces up to homeomorphism, and if X,Y are such spaces then X is embeddable into Y. Furthermore, if T is a topology on the set R finer than the Euclidean topology and the space (R,T) is separable and locally connected then the space is locally compact and homeomorphic to some space in S. In a very natural way we establish a complete classification of all these refinements T of the real line. |
| title | On real functions with graphs either connected or locally connected |
| topic | General Topology |
| url | https://arxiv.org/abs/2510.12450 |