Some theorems involving compactly generated#290
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Approved, though maybe https://topology.pi-base.org/theorems/T000070 and https://topology.pi-base.org/theorems/T000292 can be consolidated somehow. |
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Actually maybe https://topology.pi-base.org/theorems/T000070 should just be "generated by compact T2 subspaces" implies "T1": Consider a point: its intersection with every (compact T2 =>) T1 subspace is either empty or a point within a T1 space, and thus closed |
Yeah, I was wondering about that too. For the sake of discussion here, call the compactly generated properties CG1 (gen by compact subspaces), CG2, CG3 (gen by compact T2 subspaces). We have:
For example, every finite top space is compact, hence CG1, and also anticompact. In that case, T292 says: a finite T1 space is discrete. But this cannot be deduced from T70, the only finite CG3 space would be the discrete ones. In the other direction, can T70 be deduced from T292? CG3 implies CG1, but does it imply T1? |
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(our messages crossed each other :-) |
That's very good. Do you want to change T70 or I'll do it? |
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I'll give it a try and then you can review. BTW, what do you think of the last question I was asking in issue #230? |
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Added your suggestion. All self-contained, but I used wikipedia as a reference, just because we need one. (added later) Actually changed the ref back to the previous one, after updating https://math.stackexchange.com/questions/4646084/unraveling-the-various-definitions-of-k-space-or-compactly-generated-space/4647078#4647078. It's better that way, so wikipedia can be updated without circular reference. |
* fix line ending issues * fix: add ref name to mathse ref See pi-base/web#10 * Update CONTRIBUTING.md * Weak Hausdorff property (#237) * Weak Hausdorff property * Add properties for weak Hausdorff * T227: tweak the explanation * add kspace properties and theorems (#236) * add kspace properites and theorems * tweaking the definitions * Update P000141.md * Update P000140.md --------- Co-authored-by: Patrick Rabau <prabau@sbcglobal.net> * S37: example of US, non weak Hausdorff space (#241) * Fix typo in P84 * Add results to show that pseudometrizable implies 1st countable (#244) * Add results to show that pseudometrizable implies 1st countable * fix T327 id * fix typos in P144 * add mathse reference * Fix typo * Update T000327.md * Some theorems and traits involving local metrizability (#248) * More theorems about locally pseudometrizable (#250) * More theorems about locally pseudometrizable * T331 converse * Misc spaces: more precise position between T1 and T2 (#249) * Misc spaces: more precise position between T1 and T2 * remove unneeded T2 assertion * remove another unneeded T2 assumption --------- Co-authored-by: Steven Clontz <steven.clontz@gmail.com> * T328 update (#252) * Included locally Euclidean -> locally compact to get some examples of spaces which are locally metrizable but not locally Euclidean. * fixed parse error * fixed another typo * suggested edits * Use citation cff (#253) * Commit line-ending changes auto-enforced by git * Adds a CITATION.cff file to make use of the github citation button feature https://docs.github.com/en/repositories/managing-your-repositorys-settings-and-features/customizing-your-repository/about-citation-files * Update CITATION.cff --------- Co-authored-by: Steven Clontz <steven.clontz@gmail.com> * T230: first countable + US imply T2 (#257) * T230: first countable + US imply T2 * remove deduceable T1 --------- Co-authored-by: Steven Clontz <steven.clontz@gmail.com> * Update CITATION.cff (#260) * T209 generalization (#264) * T209 generalization * fix typo * T208: indiscrete + distinct points ==> no isolated point (#266) * S27 and S28 updates (#261) * S27 updates * S28 updates * fix typo * S27/P23: added reference * S31: square of one point compactification of Q (#263) * S153: Open long ray (#270) * S153: Open long ray * improve lindelof->weakly lindelof --------- Co-authored-by: Steven Clontz <steven.clontz@gmail.com> * split up theorems implying conjunctions of properties (#267) * split up theorems implying conjuctions of properties * remove proposed but out-of-syntax deprecatedBy * Update T000042.md * Update T000099.md * Update T000100.md * add converse * Delete T000171.md * Update T000176.md * Update T001042.md * Update T001176.md * fix typos * split T104 * fix uid * consolidate theorem IDs * add a few converses --------- Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> * more inclusive choice of default terminology (#271) * more inclusive choice of default terminology * Update README.md * Update README.md * Update README.md * add sequential property to arens space (#274) * T82: locally euclidean => locally arc connected (#272) * connect rothberger property and create counterexamples (#275) * Update P000144.md * Fix P144 line ending * fix MR refs * Fix more MR refs * add aliases for separation properties (#276) * improve regular to t3 (#277) * improve regular to t3 * improve t3 to t3.5 for S77 (and add t5 for P63) * add michael line S63 normality props * swap the factors to map S&S --------- Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> * add counterexample from https://math.stackexchange.com/q/3145712 (#279) * add counterexample * Update README.md * remove T0 assumption in T23 (#283) * remove T0 assumption in T23 * actually update T23 * Update T000023.md * add paracompact properties and arens space props (#282) * add paracompact properites and arens space props * Update P000050.md * Update P000146.md * misc changes * add alternate def for ultraparacompact --------- Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> * Some more properties for the Arens space (#285) * More Arens space updates * fix for scattered property * Fix for Fully normal (P34) (#289) * Fix for Fully normal (P34) * add one reference * generalize T214 from fully T4 to fully normal (#288) * Add a perfectly normal, non fully normal space (#287) * add perfectly normal non fully normal space * Update README.md * Updates to Fortissimo space (S22) (#292) * Some theorems involving compactly generated (#290) * Some theorems involving compactly generated * Example of space with P140 and not P141 * T70 to provide spaces with P141 and not P142 * Better version of T70 * change the ref for T70 * Convenience aliases for compactly generated (#293) * Updates to Single ultrafilter topology (#294) --------- Co-authored-by: James Dabbs <james.dabbs@gmail.com> Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com> Co-authored-by: Patrick Rabau <prabau@sbcglobal.net> Co-authored-by: Chris <30360237+ccaruvana@users.noreply.github.com> Co-authored-by: rschwiebert <rschwiebert@users.noreply.github.com>
2 participants
Generalized theorem T59, plus new theorems T60 and T69, related to various notions of compactly generated space.
See #230.