diff --git a/spaces/S000205/README.md b/spaces/S000205/README.md index e82d821ae6..f41d8193dc 100644 --- a/spaces/S000205/README.md +++ b/spaces/S000205/README.md @@ -6,12 +6,15 @@ refs: name: Warsaw circle on Wikipedia - doi: 10.1016/j.topol.2013.07.001 name: Milnor–Thurston homology groups of the Warsaw Circle (J. Przewocki) + - mathse: 2423154 + name: First Betti number of a Reeb graph is not greater than that of the space? --- The set $X=\{(x, \sin\frac{1}{x}):x \in (0,1]\} \cup \{(0,y):y \in [-2,1]\} \cup \{(x,-2):x\in[0,1]\} \cup \{(1,y):y \in [-2,\sin 1]\}$ with the subspace topology inherited from $\mathbb{R}^2$. It consists of {S114} together with an extra arc joining its rightmost point to the point $(0,-1)$. -See {{wikipedia:Warsaw_circle}} or section 3 of {{doi:10.1016/j.topol.2013.07.001}} +See {{wikipedia:Warsaw_circle}} or section 3 of {{doi:10.1016/j.topol.2013.07.001}}. -Note: There are a few closely related but non-isomorphic variants of the *Warsaw circle*. +Note: There are a few closely related but non-isomorphic variants of the *Warsaw circle* +(see the pictures in {{mathse:2423154}} for two of them). The description chosen here seems to be the most common. diff --git a/spaces/S000205/properties/P000089.md b/spaces/S000205/properties/P000089.md new file mode 100644 index 0000000000..0530bb9e5c --- /dev/null +++ b/spaces/S000205/properties/P000089.md @@ -0,0 +1,10 @@ +--- +space: S000205 +property: P000089 +value: true +refs: +- mathse: 4775855 + name: Warsaw circle has the fixed point property +--- + +See answer to {{mathse:4775855}}. diff --git a/spaces/S000205/properties/P000204.md b/spaces/S000205/properties/P000204.md new file mode 100644 index 0000000000..4f7f1c4879 --- /dev/null +++ b/spaces/S000205/properties/P000204.md @@ -0,0 +1,8 @@ +--- +space: S000205 +property: P000204 +value: false +--- + +Let $p \in X$. If $p \notin \{0\}\times [-1,1]$, then $X\setminus \{p\}$ is homotopy equivalent to {S114} and {S114|P36}. +Otherwise $Y = X \setminus (\{0\}\times [-1,1])$ is dense in $X \setminus \{p\}$. Hence $X \setminus \{p\}$ is connected as $Y$ is clearly (path-)connected.