Concerning semiconnected maps
| Autor: | Paul E. Long |
|---|---|
| Rok vydání: | 1969 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 21:117-118 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-1969-0236890-8 |
| Popis: | Introduction. Professor John Jones, Jr., [31, introduces a semiconnected map f: X-Y as one in which f-l preserves closed connected subsets of Y, and gives conditions under which a semiconnected map is continuous or is a homeomorphism. Theorem 1 of that paper is generalized here, and comparisons are made between semiconnected maps and other noncontinuous maps. Among the several other well-known types of noncontinuous maps only the connected map and the connectivity map will be considered. A connected map f: X-* Y is one which preserves connected subsets of X and a connectivity map f: X-+ Y is one for which the induced graph map, g: X->XX Y defined by g(x) = (x, f(x)) for each xEX, is connected. It is easy to see that if f: X-* Y is continuous, thenf is a connectivity map, and if a connectivity map, then also connected. Examples showing the reverse implications are not always valid may be found in [2]. The example f(x) = x2 from the reals into the reals (usual topology in both cases) shows that continuous maps, hence connected and connectivity maps, need not be semiconnected. Furthermore, f(x) = x from the reals (usual topology) to the reals (discrete topology) is semiconnected but not connected, hence not a connectivity nor a continuous map. Throughout, cl(A) denotes the closure of the set A. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|