From f3d2257ce8f5122d0dc85e1b15f8925b5a33324f Mon Sep 17 00:00:00 2001 From: Patrick Rabau Date: Sat, 18 Mar 2023 18:29:00 -0400 Subject: [PATCH 1/3] Misc spaces: more precise position between T1 and T2 --- spaces/S000019/properties/P000099.md | 10 ++++++++++ spaces/S000024/properties/P000099.md | 10 ++++++++++ spaces/S000097/README.md | 2 +- spaces/S000097/properties/P000100.md | 10 ++++++++++ spaces/S000165/README.md | 2 +- spaces/S000165/properties/P000143.md | 12 ++++++++++++ 6 files changed, 44 insertions(+), 2 deletions(-) create mode 100644 spaces/S000019/properties/P000099.md create mode 100644 spaces/S000024/properties/P000099.md create mode 100644 spaces/S000097/properties/P000100.md create mode 100644 spaces/S000165/properties/P000143.md diff --git a/spaces/S000019/properties/P000099.md b/spaces/S000019/properties/P000099.md new file mode 100644 index 0000000000..664ba65d02 --- /dev/null +++ b/spaces/S000019/properties/P000099.md @@ -0,0 +1,10 @@ +--- +space: S000019 +property: P000099 +value: false +refs: +- doi: 10.1007/978-1-4612-6290-9_6 + name: Counterexamples in Topology +--- + +The sequence $(n)_{n\in\omega}$ is eventually outside of every compact subset of $\mathbb R$, that is, inside of every nonempty open set in $X$. So it converges to every point of $X$. diff --git a/spaces/S000024/properties/P000099.md b/spaces/S000024/properties/P000099.md new file mode 100644 index 0000000000..4330b73b2d --- /dev/null +++ b/spaces/S000024/properties/P000099.md @@ -0,0 +1,10 @@ +--- +space: S000024 +property: P000099 +value: false +refs: +- doi: 10.1007/978-1-4612-6290-9_6 + name: Counterexamples in Topology +--- + +An infinite sequence of distinct points of $\mathbb R$ converges to both $\infty_1$ and $\infty_2$. diff --git a/spaces/S000097/README.md b/spaces/S000097/README.md index 7af658494b..eac2a7cf02 100644 --- a/spaces/S000097/README.md +++ b/spaces/S000097/README.md @@ -7,7 +7,7 @@ refs: - doi: 10.1007/978-1-4612-6290-9 name: Counterexamples in Topology --- -Let $X$ be the set of all lattice points $(i,j) \in \omega^2$ along with two extra points $x$ and $y$. Let each $(i,j)$ be open. Neighborhoods of $x$ have the form $X \setminus A$ where $A$ is any set of lattice points with at most finitely many points on each row. Neighborhoods of $y$ have the form $X \setminus B$ where $B$ is any set of lattice points selected from at most finitely many rows. +Let $X$ be the set of all lattice points $(i,j) \in \omega^2$ together with two extra points $x$ and $y$. Let each $(i,j)$ be open. Basic neighborhoods of $x$ have the form $X \setminus (A\cup\{y\})$ where $A$ is any set of lattice points with at most finitely many points on each row. Basic neighborhoods of $y$ have the form $X \setminus (B\cup\{x\})$ where $B$ is any set of lattice points selected from at most finitely many rows. Defined as counterexample #99 ("Maximal Compact Topology") in {{doi:10.1007/978-1-4612-6290-9}}. diff --git a/spaces/S000097/properties/P000100.md b/spaces/S000097/properties/P000100.md new file mode 100644 index 0000000000..54b4b0f8a9 --- /dev/null +++ b/spaces/S000097/properties/P000100.md @@ -0,0 +1,10 @@ +--- +space: S000097 +property: P000100 +value: true +refs: +- doi: 10.1007/978-1-4612-6290-9_6 + name: Counterexamples in Topology +--- + +See item #3 for space #99 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000165/README.md b/spaces/S000165/README.md index 83bb4be2c1..d87b0f7a25 100644 --- a/spaces/S000165/README.md +++ b/spaces/S000165/README.md @@ -7,4 +7,4 @@ refs: name: Answer to ""All retracts are closed" and "all compacts are closed"" --- -The one-point compactification of {S23}, see {{mo:435257}}. +The one-point compactification $X$ of {S23}. See [this answer](https://mathoverflow.net/a/435257) to {{mo:435257}}. diff --git a/spaces/S000165/properties/P000143.md b/spaces/S000165/properties/P000143.md new file mode 100644 index 0000000000..2754da86d9 --- /dev/null +++ b/spaces/S000165/properties/P000143.md @@ -0,0 +1,12 @@ +--- +space: S000165 +property: P000143 +value: false +refs: + - mo: 435257 + name: Answer to ""All retracts are closed" and "all compacts are closed"" +--- + +Let $0$ be the non-isolated point of the Arens-Fort space. The subspace $A=X\setminus\{0\}$ is homeomorphic to the one-point compactification of a discrete space (i.e., {S20}), which is compact and Hausdorff. But $A$ is not closed in $X$. + +See [this answer](https://mathoverflow.net/a/435257) to {{mo:435257}}. From 441b97a0a53b5e2823f012cacfc31fa7bc5de9d2 Mon Sep 17 00:00:00 2001 From: Steven Clontz Date: Sun, 19 Mar 2023 18:28:38 -0500 Subject: [PATCH 2/3] remove unneeded T2 assertion --- spaces/S000024/properties/P000003.md | 10 ---------- 1 file changed, 10 deletions(-) delete mode 100644 spaces/S000024/properties/P000003.md diff --git a/spaces/S000024/properties/P000003.md b/spaces/S000024/properties/P000003.md deleted file mode 100644 index 911da751d8..0000000000 --- a/spaces/S000024/properties/P000003.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000024 -property: P000003 -value: false -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -See item #3 for space #27 in {{doi:10.1007/978-1-4612-6290-9_6}}. From c4a9cbde7e9ea2f77a418aaa172023e841596307 Mon Sep 17 00:00:00 2001 From: Steven Clontz Date: Sun, 19 Mar 2023 18:36:09 -0500 Subject: [PATCH 3/3] remove another unneeded T2 assumption --- spaces/S000165/properties/P000003.md | 11 ----------- 1 file changed, 11 deletions(-) delete mode 100644 spaces/S000165/properties/P000003.md diff --git a/spaces/S000165/properties/P000003.md b/spaces/S000165/properties/P000003.md deleted file mode 100644 index 78dbc081f8..0000000000 --- a/spaces/S000165/properties/P000003.md +++ /dev/null @@ -1,11 +0,0 @@ ---- -space: S000165 -property: P000003 -value: false -refs: - - mo: 435257 - name: Answer to ""All retracts are closed" and "all compacts are closed"" ---- - -The space is a the one-point compactification of a non-{P130} space. -