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Add hereditarily connected property (#800)
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--- | ||
uid: P000196 | ||
name: Hereditarily connected | ||
aliases: | ||
- Totally ordered topology | ||
refs: | ||
- doi: 10.5186/aasfm.1977.0321 | ||
name: On ultrapseudocompact and related spaces (T. Nieminen) | ||
--- | ||
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Any of the following equivalent properties holds: | ||
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- Any subspace is connected. | ||
- The open sets are totally ordered by inclusion. | ||
- The closed sets are totally ordered by inclusion. | ||
- The specialization preorder is total. | ||
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For proof of the equivalences and further characterizations, see section 12 of {{doi:10.5186/aasfm.1977.0321}}. |
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--- | ||
space: S000187 | ||
property: P000196 | ||
value: true | ||
--- | ||
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By construction. |
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--- | ||
space: S000199 | ||
property: P000051 | ||
value: true | ||
--- | ||
Since $\omega$ is well-ordered, a nonempty set $Y\subseteq\omega$ has a least element $\alpha$. | ||
Then $\alpha$ is isolated in $Y$ since $[0,\alpha+1)\cap Y=\{\alpha\}$. |
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--- | ||
space: S000199 | ||
property: P000136 | ||
value: true | ||
--- | ||
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Every infinite subspace is homeomorphic to the space itself, and any open cover not containing $\omega$ must be an infinite set of left rays and so cannot have a finite subcover. |
4 changes: 2 additions & 2 deletions
4
spaces/S000187/properties/P000039.md → spaces/S000199/properties/P000181.md
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@@ -1,6 +1,6 @@ | ||
--- | ||
space: S000187 | ||
property: P000039 | ||
space: S000199 | ||
property: P000181 | ||
value: true | ||
--- | ||
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|
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--- | ||
space: S000199 | ||
property: P000196 | ||
value: true | ||
--- | ||
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By construction. |
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--- | ||
space: S000200 | ||
property: P000016 | ||
value: true | ||
--- | ||
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Any open cover must contain $\omega$. |
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4 changes: 2 additions & 2 deletions
4
spaces/S000187/properties/P000040.md → spaces/S000200/properties/P000181.md
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--- | ||
space: S000187 | ||
property: P000040 | ||
space: S000200 | ||
property: P000181 | ||
value: true | ||
--- | ||
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|
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--- | ||
space: S000200 | ||
property: P000196 | ||
value: true | ||
--- | ||
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||
By construction. |
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--- | ||
uid: T000546 | ||
if: | ||
P000196: true | ||
then: | ||
P000039: true | ||
--- | ||
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The open sets are totally ordered by set inclusion and thus no two nonempty ones are disjoint. |
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--- | ||
uid: T000547 | ||
if: | ||
P000196: true | ||
then: | ||
P000014: true | ||
--- | ||
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Any {P196} space is trivially {P13} (as there are no disjoint closed sets), so all subspaces are as well. |
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--- | ||
uid: T000548 | ||
if: | ||
and: | ||
- P000039: true | ||
- P000014: true | ||
then: | ||
P000196: true | ||
refs: | ||
- doi: 10.5186/aasfm.1977.0321 | ||
name: On ultrapseudocompact and related spaces (T. Nieminen) | ||
--- | ||
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See condition (10) of theorem 23 at {{doi:10.5186/aasfm.1977.0321}} (reading $T_5$ as {P14}). |
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--- | ||
uid: T000549 | ||
if: | ||
P000196: true | ||
then: | ||
P000174: true | ||
--- | ||
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The topology is totally ordered by inclusion, so the set of open neighborhoods of any point is as well. |
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--- | ||
uid: T000550 | ||
if: | ||
and: | ||
- P000193: true | ||
- P000039: true | ||
then: | ||
P000146: true | ||
--- | ||
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In a {P39} space $X$, the closure of every nonempty open set is $X$, so any open cover that admits a shrinking must contain $X$ (as otherwise each of its open sets could only contain the closure of $\varnothing$). | ||
Thus, any such open cover admits an open refinement to the partition $\{X\}$. |
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--- | ||
uid: T000551 | ||
if: | ||
and: | ||
- P000196: true | ||
- P000093: true | ||
- P000057: false | ||
then: | ||
P000114: true | ||
--- | ||
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By {P93} choose a countable open neighborhood of each point. So $X$ can be covered by a family of countable sets, forming a chain under inclusion by {P196}. And the union of a chain of countable sets has cardinality at most $\aleph_1$ (see {{mathse:342091}} for example). |
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--- | ||
uid: T000552 | ||
if: | ||
and: | ||
- P000146: true | ||
- P000036: true | ||
- P000086: true | ||
then: | ||
P000129: true | ||
--- | ||
Any clopen partition of a {P36} space $X$ must contain $X$, so every open cover admitting a clopen partition as refinement must contain $X$. Thus, the union of all open sets except for $X$ cannot equal $X$ (as that would be an open cover not containing $X$), so some points have $X$ as their only neighborhood. By homogeneity, this must then be true of all points, i.e. the space is {P129}. |
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--- | ||
uid: T000553 | ||
if: | ||
P000196: true | ||
then: | ||
P000042: true | ||
--- | ||
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Every subset of a {P196} space is {P40}, and {T38}. |
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--- | ||
uid: T000554 | ||
if: | ||
and: | ||
- P000196: true | ||
- P000016: true | ||
- P000073: true | ||
- P000090: true | ||
then: | ||
P000075: true | ||
refs: | ||
- doi: 10.1017/9781316543870 | ||
name: Spectral spaces (Dickmann, Schwartz, Tressl) | ||
--- | ||
Shown in Proposition 1.6.7 of {{doi:10.1017/9781316543870}}. | ||
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Furthermore, if the space is nonempty, its specialization order is order-isomorphic to a successor ordinal $\lambda$ and the space is homeomorphic to $\lambda$ with the left-ray topology having the collection of intervals $[0,\alpha]$ ($\alpha\in\lambda$) as a base. |
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--- | ||
uid: T000555 | ||
if: | ||
and: | ||
- P000196: true | ||
- P000176: true | ||
then: | ||
P000044: false | ||
--- | ||
$X$ has two disjoint subsets, each with at least two points, and each of the subsets is connected. |
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--- | ||
uid: T000556 | ||
if: | ||
and: | ||
- P000146: true | ||
- P000036: true | ||
then: | ||
P000020: true | ||
--- | ||
Any clopen partition of a {P36} space $X$ must contain $X$, so to admit clopen refinements every open cover must contain $X$. Thus, the union of all open sets except for $X$ cannot equal $X$ (as that would be an open cover not containing $X$), so any sequence converges to all points outside of that union (whose only neighborhood is $X$). |
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--- | ||
uid: T000557 | ||
if: | ||
and: | ||
- P000001: true | ||
- P000090: true | ||
- P000027: true | ||
then: | ||
P000057: true | ||
--- | ||
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In an {P90} space, the smallest basis is the set of smallest neighborhoods of points. | ||
When it is also {P1}, these are all distinct, so they are in bijection with the set of points. | ||
Thus, such a space has a countable basis iff it is countable. |