diff --git a/spaces/S000139/properties/P000064.md b/spaces/S000139/properties/P000064.md deleted file mode 100644 index 4c6745586f..0000000000 --- a/spaces/S000139/properties/P000064.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000139 -property: P000064 -value: true -refs: - - wikipedia: Baire_space - name: Baire space on Wikipedia ---- - -The subspace $X\setminus\{\infty\}$ is Baire (because it is locally compact and Hausdorff) and dense in $X.$ diff --git a/spaces/S000139/properties/P000206.md b/spaces/S000139/properties/P000206.md new file mode 100644 index 0000000000..fadb971900 --- /dev/null +++ b/spaces/S000139/properties/P000206.md @@ -0,0 +1,8 @@ +--- +space: S000139 +property: P000206 +value: true +--- + +If Player I picks $x_n= \infty$ in every turn, clearly $\infty \in \bigcap \{U_n:n<\omega\}$. +So suppose for some $n$, $x_n\neq \infty$, then Player II can choose $V_n$ to be homeomorphic to {S25} and the assertion follows since {S25|P206}. diff --git a/spaces/S000139/properties/P000214.md b/spaces/S000139/properties/P000214.md new file mode 100644 index 0000000000..597d11ae34 --- /dev/null +++ b/spaces/S000139/properties/P000214.md @@ -0,0 +1,7 @@ +--- +space: S000139 +property: P000214 +value: false +--- + +For $m \in \mathbb{N}$, let $S_m := \{m + \frac{1}{2}, m + \frac{1}{3}, \dots\}$. Clearly $S_m$ converges to $\infty$, but picking any countable infinite $S$ with $S \cap S_m \neq \emptyset$ for infinitely many $m$ does not.