diff --git a/spaces/S000022/README.md b/spaces/S000022/README.md index 5b50f8a6dd..92683ddd6c 100644 --- a/spaces/S000022/README.md +++ b/spaces/S000022/README.md @@ -1,19 +1,19 @@ --- uid: S000022 -name: Fortissimo Space on the Real Numbers +name: Fortissimo space on the real numbers aliases: - - Fortissimo Space + - Fortissimo space - One-point Lindelöfication of uncountable discrete space counterexamples_id: 25 refs: - doi: 10.1007/978-1-4612-6290-9 name: Counterexamples in Topology - wikipedia: Fort_space - name: Fort space + name: Fort space on Wikipedia --- Let $X=\mathbb{R}\cup\{\infty\}$. -Define $U \subseteq X$ to be open if it does not contain $\infty$ or its complement is countable. +Every point of $\mathbb R$ is isolated and the open neighborhoods of the point $\infty$ are the cocountable subsets of $X$ containing that point. This space is the one-point Lindelöfication of an uncountable discrete space. diff --git a/spaces/S000022/properties/P000008.md b/spaces/S000022/properties/P000002.md similarity index 57% rename from spaces/S000022/properties/P000008.md rename to spaces/S000022/properties/P000002.md index 2b519fdea5..7eb3990da3 100644 --- a/spaces/S000022/properties/P000008.md +++ b/spaces/S000022/properties/P000002.md @@ -1,10 +1,10 @@ --- space: S000022 -property: P000008 +property: P000002 value: true refs: - doi: 10.1007/978-1-4612-6290-9_6 name: Counterexamples in Topology --- -See item #1 for space #25 in {{doi:10.1007/978-1-4612-6290-9_6}}. +By inspection, every singleton is closed. diff --git a/spaces/S000022/properties/P000022.md b/spaces/S000022/properties/P000022.md deleted file mode 100644 index 3fb17434f6..0000000000 --- a/spaces/S000022/properties/P000022.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000022 -property: P000022 -value: false -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -See item #3 for space #25 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000022/properties/P000028.md b/spaces/S000022/properties/P000028.md deleted file mode 100644 index 4f9f6195e2..0000000000 --- a/spaces/S000022/properties/P000028.md +++ /dev/null @@ -1,12 +0,0 @@ ---- -space: S000022 -property: P000028 -value: false -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -If $\{ U_i : i \in \mathbb{N} \}$ is any family of open neighborhoods of $p$, then $X \setminus U_i$ is countable for all $i$. Then $X \setminus \bigcap_{i \in \mathbb{N}} U_i$ is also countable, meaning that $\bigcap_{i \in \mathbb{N}} U_i$ is uncountable. For any $x \in \bigcap_{i \in \mathbb{N}} U_i$ distinct from $p$ the set $X \setminus \{ x \}$ is an open neighborhood of $p$ which does not have any $U_i$ as a subset. - -See item #1 for space #25 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000022/properties/P000050.md b/spaces/S000022/properties/P000050.md deleted file mode 100644 index 9fc048673b..0000000000 --- a/spaces/S000022/properties/P000050.md +++ /dev/null @@ -1,12 +0,0 @@ ---- -space: S000022 -property: P000050 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -Every singleton $\{x\}$ with $x\in\mathbb R$ is clopen. And every neighborhood of $p$ is of the form $X\setminus A$ for some countable $A\subseteq\mathbb R$, hence is also clopen. - -Asserted in the General Reference Chart for space #25 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000022/properties/P000132.md b/spaces/S000022/properties/P000132.md deleted file mode 100644 index 47414439f4..0000000000 --- a/spaces/S000022/properties/P000132.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000022 -property: P000132 -value: false -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -$\{p\}$ is closed. Let $U_n = X \setminus A_n$ be an open set containing $p$. Then $A_n$ is countable and a countable intersection $\bigcap_n U_n = X \setminus \bigcup_n A_n$ is uncountable, so $\{p\}$ is not a $G_\delta$.