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On the metrizability of the bundle space
| Content Provider | Semantic Scholar |
|---|---|
| Author | Etter, D. O. Griffin, John S. |
| Copyright Year | 1954 |
| Abstract | It has been shown by Yu. M. Smirnov (see [l, p. 13, Theorem 3]) that a Hausdorff space X is metrizable if and only if X is paracompact and has an open cover each of whose members is metrizable. Using this, we prove: If {X, B, ir, Y, V, 0, 67} is a fibre bundle (see [2, p. 7]) whose base space B and fibre Y are metrizable, then the bundle space X is also metrizable. First, if 73 and Y are metrizable, then X is Hausdorff and has an open cover each of whose members is metrizable, namely {ir-1(U)\ UEV}, since if UEV then UXY, and hence tt~\U), is metrizable. Second, X is paracompact, for let Q be any open cover for X. Let (j be a locally finite refinement of TJ, and let W be a closure refinement of Qwhich is also locally finite; define\:W—>Qa function such that, for each W, if WEW then W'EHW). Let for each VE$ |
| Starting Page | 466 |
| Ending Page | 467 |
| Page Count | 2 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1954-0063023-2 |
| Volume Number | 5 |
| Alternate Webpage(s) | https://www.ams.org/journals/proc/1954-005-03/S0002-9939-1954-0063023-2/S0002-9939-1954-0063023-2.pdf |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1954-005-03/S0002-9939-1954-0063023-2/S0002-9939-1954-0063023-2.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1954-0063023-2 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
