diff --git a/spaces/S000140/README.md b/spaces/S000140/README.md index 2a0b752e3..4f4f5eadd 100644 --- a/spaces/S000140/README.md +++ b/spaces/S000140/README.md @@ -1,6 +1,6 @@ --- uid: S000140 -name: Real numbers extended by a point with co-countable open neighborhoods +name: $\mathbb{R}$ extended by a point with co-countable open neighborhoods refs: - mathse: 4850979 name: Answer to 'Radial/pseudoradial implies Fréchet-Urysohn/sequential for locally countable spaces' @@ -8,6 +8,6 @@ refs: name: What are the compactness properties of $\mathbb R$, extended by a point with co-countable open neighborhoods? --- -Let $X=\mathbb R\cup \{\infty\}$, with $\mathbb R$ having the Euclidean topology and open in $X$, and open neighborhoods of $\infty$ given by sets of the form $U\cup\{\infty\}$, where $U\subseteq \mathbb R$ is co-countable and open in the Euclidean topology. +Let $X=\mathbb R\cup \{\infty\}$. Let $\mathbb R$ be open in $X$ and have the topology of {S25}, and let open neighborhoods of $\infty$ be given by sets of the form $U\cup\{\infty\}$, where $U\subseteq \mathbb R$ is co-countable and open in the Euclidean topology. Constructed in {{mathse:4850979}} as an example of a space that is {P173} and {P81}, yet fails to be {P79}, yielding a counterexample to a natural analogue of {T211}. Elaborated on in {{mathse:4854178}}. diff --git a/spaces/S000140/properties/P000073.md b/spaces/S000140/properties/P000073.md new file mode 100644 index 000000000..32d0aa921 --- /dev/null +++ b/spaces/S000140/properties/P000073.md @@ -0,0 +1,11 @@ +--- +space: S000140 +property: P000073 +value: true +--- + +Let $S \subseteq X$ be a nonempty irreducible ({P39}) set. $\mathbb R$ +is open in $X$ and so $S \cap \mathbb R$ is also irreducible. Since {S25|P73} and {S25|P2}, we know that +$S \cap \mathbb R$ is either empty or a singleton. Thus $S$ has at most two +points. Now note that {S140|P2}, and so if $S$ had two points, then $S$ +would be homeomorphic to {S1}. But {S1|P39}, and so $S$ is a singleton. diff --git a/spaces/S000140/properties/P000189.md b/spaces/S000140/properties/P000189.md new file mode 100644 index 000000000..229173f74 --- /dev/null +++ b/spaces/S000140/properties/P000189.md @@ -0,0 +1,7 @@ +--- +space: S000140 +property: P000189 +value: true +--- + +$\mathbb R$ is a dense subset of $X$ and {S25|P189}. diff --git a/spaces/S000140/properties/P000204.md b/spaces/S000140/properties/P000204.md new file mode 100644 index 000000000..3fa9b85e0 --- /dev/null +++ b/spaces/S000140/properties/P000204.md @@ -0,0 +1,13 @@ +--- +space: S000140 +property: P000204 +value: false +--- + +The point $\infty$ is not a cut point, since $X \setminus \{\infty\}=\mathbb R$ is connected. +Now, let $x \in \mathbb{R}$ and consider a map $f$ +from $X \setminus \{x\}$ to {S1}. Then the intervals $(-\infty, x)$ and +$(x, \infty)$ in $\mathbb{R}$ are connected and thus get sent to a +single point. Thus $f$ factors through the quotient space of $X$ +where $(-\infty, x)$ and $(x, \infty)$ become points. This space is +{S11} and {S11|P36}; thus $f$ is constant, which proves the assertion. diff --git a/spaces/S000140/properties/P000210.md b/spaces/S000140/properties/P000210.md new file mode 100644 index 000000000..2bd2921b0 --- /dev/null +++ b/spaces/S000140/properties/P000210.md @@ -0,0 +1,10 @@ +--- +space: S000140 +property: P000210 +value: true +refs: +- mathse: 5126685 + name: What properties hold for $\mathbb{R}$ extended by a point with co-countable open neighborhoods? +--- + +See {{mathse:5126685}}. diff --git a/spaces/S000140/properties/P000219.md b/spaces/S000140/properties/P000219.md new file mode 100644 index 000000000..44a0c0bf2 --- /dev/null +++ b/spaces/S000140/properties/P000219.md @@ -0,0 +1,7 @@ +--- +space: S000140 +property: P000219 +value: false +--- + +{S140|P65} and has a subspace homeomorphic to {S25}.