diff --git a/spaces/S000171/README.md b/spaces/S000171/README.md index eef348b0c4..876a2f80e5 100644 --- a/spaces/S000171/README.md +++ b/spaces/S000171/README.md @@ -1,10 +1,20 @@ --- uid: S000171 -name: Brian's Example +name: Brian's stack of Bernstein sets refs: - - mo: 416331 - name: Example of an uncountable scattered space with some properties + - mo: 416752 + name: Answer to "Example of an uncountable scattered space with some properties" + - wikipedia: Bernstein_set + name: Bernstein set on Wikipedia --- -In {{mo:416331}} Will Brian provides an example in ZFC for an uncountable, Hausdorff, +Let $X = \mathbb{R}$ and let $(X_i)_{i\in \omega}$ be a countably infinite partition of $X$ into +[Bernstein sets](https://en.wikipedia.org/wiki/Bernstein_set) (for the Euclidean topology). +Given $x \in X$, let $n \in \omega$ be such that $x \in X_n$. +Then basic open neighbourhoods of $x$ are given by the sets $\{x\} \cup (U \cap \bigcup_{i