From 487fc940366a2b76312e422f528323123bb18dff Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Mon, 11 May 2026 10:45:50 +0800 Subject: [PATCH 1/2] Complete Radial interval topology --- spaces/S000135/properties/P000030.md | 10 ---------- spaces/S000135/properties/P000043.md | 10 ---------- spaces/S000135/properties/P000118.md | 29 ++++++++++++++++++++++++++++ spaces/S000135/properties/P000132.md | 18 ----------------- spaces/S000135/properties/P000240.md | 8 ++++++++ 5 files changed, 37 insertions(+), 38 deletions(-) delete mode 100644 spaces/S000135/properties/P000030.md delete mode 100644 spaces/S000135/properties/P000043.md create mode 100644 spaces/S000135/properties/P000118.md delete mode 100644 spaces/S000135/properties/P000132.md create mode 100644 spaces/S000135/properties/P000240.md diff --git a/spaces/S000135/properties/P000030.md b/spaces/S000135/properties/P000030.md deleted file mode 100644 index d3a2ca6d04..0000000000 --- a/spaces/S000135/properties/P000030.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000135 -property: P000030 -value: true -refs: -- zb: "0386.54001" - name: Counterexamples in Topology ---- - -See item #5 for space #141 in {{zb:0386.54001}} diff --git a/spaces/S000135/properties/P000043.md b/spaces/S000135/properties/P000043.md deleted file mode 100644 index 0ff6d75c12..0000000000 --- a/spaces/S000135/properties/P000043.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000135 -property: P000043 -value: true -refs: -- zb: "0386.54001" - name: Counterexamples in Topology ---- - -A basic neighbourhood is either a Euclidean segment or the union of a family of such segments with nonempty intersection. diff --git a/spaces/S000135/properties/P000118.md b/spaces/S000135/properties/P000118.md new file mode 100644 index 0000000000..eb04bef790 --- /dev/null +++ b/spaces/S000135/properties/P000118.md @@ -0,0 +1,29 @@ +--- +space: S000135 +property: P000118 +value: false +--- + +By hereditary, we show that a variant of {S131}, $S_{\omega_1} = (\omega_1 \times \omega) \cup \{\infty\}$ (the sequential fan with $\omega_1$-many spines) is not {P118}. + +Let $\mathcal N = \bigcup_{i \in \mathbb N} \mathcal N_i$ be a $k$-network of $S_{\omega_1}$, where each $\mathcal N_i$ is locally finite. + +Define $\mathcal A_i = \left\{ A \in \mathcal N_i \mid \infty \in \operatorname{cl}(A) \right\}$, then $\mathcal A_i$ is finite by locally finiteness of $\mathcal N_i$. + +Therefore, $\mathcal A := \left\{ A \in \mathcal N \mid \infty \in \operatorname{cl}(A) \right\} = \bigcup_{i \in \mathbb N} \mathcal A_i$ is countable. + +--- +For any subset $A \subset S_{\omega_1}$, define \[ \operatorname{supp}(A) = \left\{ m \in \omega_1 \mid \exists n \in \omega, (m, n) \in A \right\}. \] + +Split $\mathcal A$ into $\mathcal F = \left\{ A \in \mathcal A \mid \left| \operatorname{supp}(A) \right| < \aleph_0 \right\}$ and $\mathcal I = \left\{ A \in \mathcal A \mid \left| \operatorname{supp}(A) \right| \geq \aleph_0 \right\}$, define $V = \bigcup_{A \in \mathcal F} \operatorname{supp}(A) \subseteq \omega_1$, which is countable, hence there exists $v \in V^c$. + +Now list $\mathcal I = \left\{ I_0, I_1, \dots \right\}$, for each $i$, take $(m_i, n_i) \in I_i$. Since $\operatorname{supp}(I_i) \geq \aleph_0$, we can force all $m_i$'s and $v$ are distinct, hence $F = \left\{ (m_i, n_i) \mid i \in \mathbb N \right\}$ are closed. + +--- +Define $K := (\{v\} \times \omega) \cup \{\infty\} \subseteq F^c =: U$, suppose $K \subseteq A_1 \cup A_2 \cup \dots \cup A_p \subseteq U$. + +* If $A_i = I_j$ for some $j$, $(m_j, n_j) \in I_j = A_i \subseteq U = F^c$, a contradiction. +* If $A_i \in \mathcal F$, then $A_i \cap K \subseteq \{\infty\}$. +* If $A_i \notin \mathcal A$, then $\infty \notin \operatorname{cl}(A)$, meaning that there exists $n$ such that $A_i \cap K \subseteq \left\{ v \right\} \times [0, n)$. + +Finite such sets cannot cover the compact set $K$. diff --git a/spaces/S000135/properties/P000132.md b/spaces/S000135/properties/P000132.md deleted file mode 100644 index e5617c5922..0000000000 --- a/spaces/S000135/properties/P000132.md +++ /dev/null @@ -1,18 +0,0 @@ ---- -space: S000135 -property: P000132 -value: true -refs: -- zb: "0386.54001" - name: Counterexamples in Topology ---- - -Since $X\setminus\{(0,0)\}$ is homeomorphic to the disjoint union of copies of the real line -and {S25|P132}, every open set in $X$ missing the origin is an $F_\sigma$. -It remains to show that the origin has a basis of open $F_\sigma$ neighborhoods. -Consider $U=\bigcup\{(-\varepsilon_\theta,\varepsilon_\theta)p_\theta: 0\leq \theta< \pi\}$, with $\varepsilon_\theta>0$ -and $p_\theta=(\cos\theta,\sin\theta)$ for each angle $0\leq \theta < \pi$. Then the set -$V:=\bigcup\{[-\varepsilon_\theta,\varepsilon_\theta]p_\theta: 0\leq \theta< \pi\}$ is closed -and $U=\bigcup_{n=2}^\infty (1-1/n)V$. - -Therefore every open subset of $X$ can be represented as a union of two $F_\sigma$ sets, so it is $F_\sigma$ as well. diff --git a/spaces/S000135/properties/P000240.md b/spaces/S000135/properties/P000240.md new file mode 100644 index 0000000000..3759150794 --- /dev/null +++ b/spaces/S000135/properties/P000240.md @@ -0,0 +1,8 @@ +--- +space: S000135 +property: P000240 +value: true +--- + +It is a $1$-dimensional {P240} with $X_0 = \left\{ x \in X \mid \left\|x\right\|_2 \in \mathbb N \right\} \subset X_1 = X$, +where the sets $1$-cells are segments connecting $(n \cos \theta, n \sin \theta)$ and $((n + 1) \cos \theta, (n + 1) \sin \theta)$ for each $n \in \mathbb N$ and $0 \leq \theta < 2 \pi$. From 4fb85043a5e0b0eb0950282919735b2fc149aac4 Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Tue, 12 May 2026 15:53:45 +0800 Subject: [PATCH 2/2] =?UTF-8?q?update=20s135|=C2=ACp118?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- spaces/S000135/properties/P000118.md | 27 ++++----------------------- 1 file changed, 4 insertions(+), 23 deletions(-) diff --git a/spaces/S000135/properties/P000118.md b/spaces/S000135/properties/P000118.md index eb04bef790..ceb064391f 100644 --- a/spaces/S000135/properties/P000118.md +++ b/spaces/S000135/properties/P000118.md @@ -2,28 +2,9 @@ space: S000135 property: P000118 value: false +refs: + - mathse: 5136509 + name: Is $\mathbb R^2$ with the radial interval topology an $\aleph$-space or a $\sigma$-space? --- -By hereditary, we show that a variant of {S131}, $S_{\omega_1} = (\omega_1 \times \omega) \cup \{\infty\}$ (the sequential fan with $\omega_1$-many spines) is not {P118}. - -Let $\mathcal N = \bigcup_{i \in \mathbb N} \mathcal N_i$ be a $k$-network of $S_{\omega_1}$, where each $\mathcal N_i$ is locally finite. - -Define $\mathcal A_i = \left\{ A \in \mathcal N_i \mid \infty \in \operatorname{cl}(A) \right\}$, then $\mathcal A_i$ is finite by locally finiteness of $\mathcal N_i$. - -Therefore, $\mathcal A := \left\{ A \in \mathcal N \mid \infty \in \operatorname{cl}(A) \right\} = \bigcup_{i \in \mathbb N} \mathcal A_i$ is countable. - ---- -For any subset $A \subset S_{\omega_1}$, define \[ \operatorname{supp}(A) = \left\{ m \in \omega_1 \mid \exists n \in \omega, (m, n) \in A \right\}. \] - -Split $\mathcal A$ into $\mathcal F = \left\{ A \in \mathcal A \mid \left| \operatorname{supp}(A) \right| < \aleph_0 \right\}$ and $\mathcal I = \left\{ A \in \mathcal A \mid \left| \operatorname{supp}(A) \right| \geq \aleph_0 \right\}$, define $V = \bigcup_{A \in \mathcal F} \operatorname{supp}(A) \subseteq \omega_1$, which is countable, hence there exists $v \in V^c$. - -Now list $\mathcal I = \left\{ I_0, I_1, \dots \right\}$, for each $i$, take $(m_i, n_i) \in I_i$. Since $\operatorname{supp}(I_i) \geq \aleph_0$, we can force all $m_i$'s and $v$ are distinct, hence $F = \left\{ (m_i, n_i) \mid i \in \mathbb N \right\}$ are closed. - ---- -Define $K := (\{v\} \times \omega) \cup \{\infty\} \subseteq F^c =: U$, suppose $K \subseteq A_1 \cup A_2 \cup \dots \cup A_p \subseteq U$. - -* If $A_i = I_j$ for some $j$, $(m_j, n_j) \in I_j = A_i \subseteq U = F^c$, a contradiction. -* If $A_i \in \mathcal F$, then $A_i \cap K \subseteq \{\infty\}$. -* If $A_i \notin \mathcal A$, then $\infty \notin \operatorname{cl}(A)$, meaning that there exists $n$ such that $A_i \cap K \subseteq \left\{ v \right\} \times [0, n)$. - -Finite such sets cannot cover the compact set $K$. +See {{mathse:5136509}}.