diff --git a/properties/P000122.md b/properties/P000122.md index c7ff4ba6f..12b1c8f6c 100644 --- a/properties/P000122.md +++ b/properties/P000122.md @@ -21,3 +21,9 @@ $\mathbb R^0=\{0\}$ is an isolated point. The closely related {P123} property is defined on page 316 of {{zb:0951.54001}} and on page 38 of {{mr:2766102}}. + +---- +#### Meta-properties + +- This property is preserved by finite products. +- This property is preserved by arbitrary disjoint unions. diff --git a/properties/P000184.md b/properties/P000184.md index b7c2d7537..b6d5326c5 100644 --- a/properties/P000184.md +++ b/properties/P000184.md @@ -16,3 +16,4 @@ Per Theorems 1.1.2 and 1.11.4 of {{zb:0401.54029}}, this is equivalent to being #### Meta-properties - This property is hereditary. +- This property is preserved by finite products. diff --git a/spaces/S000224/README.md b/spaces/S000224/README.md new file mode 100644 index 000000000..a5479a26c --- /dev/null +++ b/spaces/S000224/README.md @@ -0,0 +1,14 @@ +--- +uid: S000224 +name: Torus +refs: + - wikipedia: Torus + name: Torus on Wikipedia +--- + +The product space {S170} $\times$ {S170}. + +Equivalently, the quotient of +$[0,1]^2$ obtained by identifying $(x,0)\sim(x,1)$ and $(0,y)\sim(1,y)$. + +Homer Simpson's favourite food. diff --git a/spaces/S000224/properties/P000016.md b/spaces/S000224/properties/P000016.md new file mode 100644 index 000000000..8a2f12203 --- /dev/null +++ b/spaces/S000224/properties/P000016.md @@ -0,0 +1,7 @@ +--- +space: S000224 +property: P000016 +value: true +--- + +$X$ is the square of {S170} and {S170|P16}. diff --git a/spaces/S000224/properties/P000037.md b/spaces/S000224/properties/P000037.md new file mode 100644 index 000000000..715753e9e --- /dev/null +++ b/spaces/S000224/properties/P000037.md @@ -0,0 +1,7 @@ +--- +space: S000224 +property: P000037 +value: true +--- + +$X$ is the square of {S170} and {S170|P37}. diff --git a/spaces/S000224/properties/P000065.md b/spaces/S000224/properties/P000065.md new file mode 100644 index 000000000..47523cfe0 --- /dev/null +++ b/spaces/S000224/properties/P000065.md @@ -0,0 +1,7 @@ +--- +space: S000224 +property: P000065 +value: true +--- + +Immediate from the definitions. diff --git a/spaces/S000224/properties/P000089.md b/spaces/S000224/properties/P000089.md new file mode 100644 index 000000000..b7467451c --- /dev/null +++ b/spaces/S000224/properties/P000089.md @@ -0,0 +1,7 @@ +--- +space: S000224 +property: P000089 +value: false +--- + +The the function $(z,w)\mapsto(-z,w)$ is continuous self-map with no fixed points. diff --git a/spaces/S000224/properties/P000122.md b/spaces/S000224/properties/P000122.md new file mode 100644 index 000000000..da90d785c --- /dev/null +++ b/spaces/S000224/properties/P000122.md @@ -0,0 +1,7 @@ +--- +space: S000224 +property: P000122 +value: true +--- + +$X$ is the square of {S170} and {S170|P122}. diff --git a/spaces/S000224/properties/P000155.md b/spaces/S000224/properties/P000155.md new file mode 100644 index 000000000..75ec9ce12 --- /dev/null +++ b/spaces/S000224/properties/P000155.md @@ -0,0 +1,7 @@ +--- +space: S000224 +property: P000155 +value: false +--- + +Since {S170|P155}, $X$ is locally homeomorph to {S176}, which is not homeomorph to {S25}. diff --git a/spaces/S000224/properties/P000184.md b/spaces/S000224/properties/P000184.md new file mode 100644 index 000000000..07785126a --- /dev/null +++ b/spaces/S000224/properties/P000184.md @@ -0,0 +1,10 @@ +--- +space: S000224 +property: P000184 +value: true +refs: + - wikipedia: Torus + name: Torus on Wikipedia +--- + +$X$ is the square of {S170} and {S170|P16}. diff --git a/spaces/S000224/properties/P000200.md b/spaces/S000224/properties/P000200.md new file mode 100644 index 000000000..b07681b24 --- /dev/null +++ b/spaces/S000224/properties/P000200.md @@ -0,0 +1,10 @@ +--- +space: S000224 +property: P000200 +value: false +refs: + - zb: "1044.55001" + name: Algebraic Topology (Hatcher) +--- + +We have $\pi_1(X)\cong\pi_1(S^1\times S^1) \equiv \pi_1(S^1)\times \pi_1(S^1)\cong \mathbb{Z}\times\mathbb{Z}$. \ No newline at end of file