diff --git a/spaces/S000008/properties/P000205.md b/spaces/S000008/properties/P000205.md deleted file mode 100644 index 8899a5c371..0000000000 --- a/spaces/S000008/properties/P000205.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000008 -property: P000205 -value: false ---- - -If $x\neq p$ then any bijection $f:X\setminus\{x\}\to X$ with $f(p) = p$ is a homeomorphism, and {S8|P36}, so no $x\in X\setminus \{p\}$ is a cut point. diff --git a/spaces/S000009/properties/P000205.md b/spaces/S000009/properties/P000205.md deleted file mode 100644 index d0876ebf94..0000000000 --- a/spaces/S000009/properties/P000205.md +++ /dev/null @@ -1,8 +0,0 @@ ---- -space: S000009 -property: P000205 -value: false ---- - -Removing any point other than the particular point from $X$ results in a space homeomorphic to $X$ -and {S9|P36}. diff --git a/theorems/T000928.md b/theorems/T000928.md index ae3a8be96c..05f73c8f25 100644 --- a/theorems/T000928.md +++ b/theorems/T000928.md @@ -1,15 +1,16 @@ --- uid: T000928 if: - and: - - P000045: true - - P000175: true + P000045: true then: - P000086: false + P000205: false refs: - mathse: 5146777 name: Answer to "A connected space cannot have more than one dispersion point" --- -If $|X|\ge 3$, $X$ can have at most one dispersion point. -See Proposition 1 in {{mathse:5146777}}. +Let $p$ be a dispersion point of $X$. If $|X|\ge 2$ and $q$ is a point distinct from $p$, +the set $X\setminus\{q\}$ is connected by Proposition 2 in {{mathse:5146777}}, +so $q$ is not a cut point. +And if $|X|\le 1$, $X$ has no cut point since {T558}. +In both cases $X$ is not a {P205}.