diff --git a/spaces/S000060/properties/P000032.md b/spaces/S000060/properties/P000032.md deleted file mode 100644 index 5bd56ca4e..000000000 --- a/spaces/S000060/properties/P000032.md +++ /dev/null @@ -1,16 +0,0 @@ ---- -space: S000060 -property: P000032 -value: false ---- - -Take any countable partition $\{A_n:n<\omega\}$ of $\mathbb R\setminus\mathbb Q$ into Euclidean-dense subsets. -Then $\mathscr U:=\{A_n\cup\mathbb Q: n<\omega\}$ is -a countable open cover of $X$. Let $\mathscr V$ be an open refinement of $\mathscr U$. -Let $U$ be any neighbourhood of $x\in\mathbb R\setminus\mathbb Q$. The closure of $U$ contains an -interval around $x$, hence for each $n<\omega$ there exists -$a_n\in A_n\cap \overline{U}$. -If $a_n\in V\in\mathscr V$, the open set $V$ intersects $U$. -Note that each $a_n$ is contained in a different element of $\mathscr V$. -Therefore $U$ intersects infinitely many elements of $\mathscr V$, -proving $\mathscr V$ is not locally finite. diff --git a/spaces/S000070/properties/P000032.md b/spaces/S000070/properties/P000032.md deleted file mode 100644 index 1a7131dd9..000000000 --- a/spaces/S000070/properties/P000032.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000070 -property: P000032 -value: false -refs: -- zb: "0386.54001" - name: Counterexamples in Topology ---- - -See item #8 for space #78 in {{zb:0386.54001}}. diff --git a/spaces/S000074/properties/P000032.md b/spaces/S000074/properties/P000032.md deleted file mode 100644 index 380aabcb6..000000000 --- a/spaces/S000074/properties/P000032.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000074 -property: P000032 -value: false -refs: -- zb: "0386.54001" - name: Counterexamples in Topology ---- - -See item #7 for space #82 in {{zb:0386.54001}}. diff --git a/theorems/T000838.md b/theorems/T000838.md new file mode 100644 index 000000000..31492b7a6 --- /dev/null +++ b/theorems/T000838.md @@ -0,0 +1,18 @@ +--- +uid: T000838 +if: + and: + - P000026: true + - P000227: true +then: + P000032: false +refs: + - zb: "0684.54001" + name: General Topology (Engelking, 1989) + - zb: "1116.54006" + name: Normality and countable paracompactness of hyperspaces of ordinals. (Kemoto, Nobuyuki) +--- + +See Exercise 5.2.C(b) in {{zb:0684.54001}}. + +See also Claim in {{zb:1116.54006}}.