Classic problems III. (English) Zbl 1454.54003
Summary: This is a survey of four problems that are “classics” in many different senses of the word, and of several related problems associated with each one. The numbering of the Classic Problems picks up where that of a similar article left off about four decades ago:
Several related problems are given for each classic problem. Consistency results are summarized, and there is a discussion of each problem that explains various implications among the related problems and justifies calling certain problems equivalent. For each classic problem, an appendix goes deeper into some implications and/or includes reminiscences. There is a purely set-theoretic problem related to Classic Problem IX. Call a filter on a set D nowhere maximal if it does not trace an ultrafilter on any subset of \(D\). Related Problem D. Is every free ultrafilter the join of two nowhere maximal filters? It is shown that the special case of ultrafilters on \(\omega\) is actually equivalent to Classic Problem IX.
- IX.
- Is every point of \({\omega}^\ast\) a butterfly point?
- X.
- Is there a nonmetrizable perfectly normal, locally connected continuum?
- XI.
- Is there a normal space with a \(\sigma\)-disjoint base that is not paracompact?
- XII.
- Is there a regular symmetrizable space with a non-\(G_{\delta}\) point?
Several related problems are given for each classic problem. Consistency results are summarized, and there is a discussion of each problem that explains various implications among the related problems and justifies calling certain problems equivalent. For each classic problem, an appendix goes deeper into some implications and/or includes reminiscences. There is a purely set-theoretic problem related to Classic Problem IX. Call a filter on a set D nowhere maximal if it does not trace an ultrafilter on any subset of \(D\). Related Problem D. Is every free ultrafilter the join of two nowhere maximal filters? It is shown that the special case of ultrafilters on \(\omega\) is actually equivalent to Classic Problem IX.
MSC:
| 54-02 | Research exposition (monographs, survey articles) pertaining to general topology |
| 54A35 | Consistency and independence results in general topology |
| 54D05 | Connected and locally connected spaces (general aspects) |
| 54D15 | Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) |
| 54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |
| 54D30 | Compactness |
| 54D45 | Local compactness, \(\sigma\)-compactness |
| 54E99 | Topological spaces with richer structures |
| 01A70 | Biographies, obituaries, personalia, bibliographies |
| 03E99 | Set theory |
| 46L35 | Classifications of \(C^*\)-algebras |
| 54D80 | Special constructions of topological spaces (spaces of ultrafilters, etc.) |
| 54E35 | Metric spaces, metrizability |
| 54F15 | Continua and generalizations |
Keywords:
butterfly point; ultrafilter; perfectly normal \(( T_6)\); continuum; metrizable; locally compact; locally connected; \(\sigma \)-disjoint; Dowker; paracompact; countably metacompact; symmetrizableReferences:
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