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| Content Provider | Springer Nature Link |
|---|---|
| Author | gura, Tsuguri Shakhmatov, Dmitri |
| Copyright Year | 1997 |
| Abstract | We prove that: (i) a pathwise connected, Hausdorff space which has a continuous selection is homeomorphic to one of the following four spaces: singleton, [0,1), [0,1] or the long lineL, (ii) a locally connected (Hausdorff) space which has a continuous selection must be orderable, and (iii) an infinite connected, Hausdorff space has exactly two continuous selections if and only if it is compact and orderable. We use these results to give various characterizations of intervals via continuous selections. For instance, (iv) a topological spaceX is homeomorphic to [0,1] if (and only if)X is infinite, separable, connected, Hausdorff space and has exactly two continuous selections, and (v) a topological spaceX is homeomorphic to [0,1) if (and only if) one of the following equivalent conditions holds: (a)X is infinite, Hausdorff, separable, pathwise connected and has exactly one continuous selection; (b)X is infinite, separable, locally connected and has exactly one continuous selection; (c)X is infinite, metric, locally connected and has exactly one continuous selection. Three examples are exhibited which demonstrate the necessity of various assumptions in our results. |
| Starting Page | 317 |
| Ending Page | 328 |
| Page Count | 12 |
| File Format | |
| ISSN | 0009725X |
| Journal | Rendiconti del Circolo Matematico di Palermo |
| Volume Number | 46 |
| Issue Number | 2 |
| e-ISSN | 19734409 |
| Language | English |
| Publisher | Springer-Verlag |
| Publisher Date | 1997-01-01 |
| Publisher Place | Milan |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Set-valued map selection continuous selection hyperspace Vietoris topology ordered space linear order interval closed interval connected space locally connected space compact space pathwise connected space arcwise connected space Selections Hyperspaces Set-valued maps Connected and locally connected spaces Compactness Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces Continua and generalizations Mathematics Algebra Geometry Analysis Applications of Mathematics |
| Content Type | Text |
| Resource Type | Article |
| Subject | Mathematics |
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