diff --git a/spaces/S000066/README.md b/spaces/S000066/README.md index 7546cd88c1..de00704cd9 100644 --- a/spaces/S000066/README.md +++ b/spaces/S000066/README.md @@ -3,14 +3,17 @@ uid: S000066 name: Double origin plane counterexamples_id: 74 refs: -- doi: 10.1007/978-1-4612-6290-9 +- zb: "0386.54001" name: Counterexamples in Topology - wikipedia: Double_origin_topology name: Double origin topology on Wikipedia --- -Let $X$ consist of the set of points of the plane $\mathbb R^{2}$ together with an additional point $0^{\ast}$. Neighborhoods of points other than the origin $0$ and the point $0^{\ast}$ are the usual open sets of $R^{2} - 0$; as a basis of neighborhoods of $0$ and $0^{\ast}$, we take $V_{n}(0) = \{(x,y):x^{2} + y^{2} < \frac{1}{n^2}, y > 0\} \cup \{0\}$ and $V_{n}(0^{\ast}) = \{(x,y):x^{2} + y^{2} < \frac{1}{n^2}, y < 0\} \cup \{0^{\ast}\}$. +Let $X$ consist of the set of points of the plane $\mathbb R^{2}$ together with an additional point $0^{\ast}$. +Neighborhoods of points other than the origin $0$ and the point $0^{\ast}$ are the usual open sets of $R^{2}\setminus \{0\}$; +as a basis of neighborhoods of $0$ and $0^{\ast}$, we take $V_{n}(0) = \{(x,y):x^{2} + y^{2} < \frac{1}{n^2},\, y > 0\} \cup \{0\}$ +and $V_{n}(0^{\ast}) = \{(x,y):x^{2} + y^{2} < \frac{1}{n^2},\, y < 0\} \cup \{0^{\ast}\}$. Defined as counterexample #74 ("Double Origin Topology") -in {{doi:10.1007/978-1-4612-6290-9}}. Note that this is *not* a copy +in {{zb:0386.54001}}. Note that this is *not* a copy of {S176} with the origin doubled in the sense of {S83}. diff --git a/spaces/S000066/properties/P000003.md b/spaces/S000066/properties/P000003.md index 9ef6aa04c8..2857ff8903 100644 --- a/spaces/S000066/properties/P000003.md +++ b/spaces/S000066/properties/P000003.md @@ -3,19 +3,10 @@ space: S000066 property: P000003 value: true refs: -- doi: 10.1007/978-1-4612-6290-9_6 +- zb: "0386.54001" name: Counterexamples in Topology --- -For any points $x , y$ other than $0$ and $0^*$ there are neighborhoods $U$ and $V$ separating $x$ and $y$, since {S176} is {P3}. For $x = 0$ and $y = 0^*$, +For any points $x , y\in\mathbb R^2\setminus\{0\}$ there are neighborhoods $U$ and $V$ separating $x$ and $y$, since {S176|P3}. +For $x = 0$ and $y = 0^*$ take $U=V_1(0)$ and $V=V_1(0^*)$. -$U = \{ (x, y) \in \mathbb R : x^2 + y^2 < 1, y > 0 \} \cup \{ 0 \}$ - -and - -$V = \{ ( x, y) \in \mathbb R : x^2 + y^2 < 1, y < 0 \} \cup \{ 0^* \}$ - -are open sets that separate $x$ and $y$. - -Asserted in the General Reference Chart for space #74 in -{{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000066/properties/P000004.md b/spaces/S000066/properties/P000004.md index a7e8c9a825..4d42b515f4 100644 --- a/spaces/S000066/properties/P000004.md +++ b/spaces/S000066/properties/P000004.md @@ -3,8 +3,8 @@ space: S000066 property: P000004 value: false refs: -- doi: 10.1007/978-1-4612-6290-9_6 +- zb: "0386.54001" name: Counterexamples in Topology --- -See item #1 for space #74 in {{doi:10.1007/978-1-4612-6290-9_6}}. +See item #1 for space #74 in {{zb:0386.54001}}. diff --git a/spaces/S000066/properties/P000010.md b/spaces/S000066/properties/P000010.md index a1a709bbf6..d743eecabd 100644 --- a/spaces/S000066/properties/P000010.md +++ b/spaces/S000066/properties/P000010.md @@ -3,9 +3,10 @@ space: S000066 property: P000010 value: true refs: -- doi: 10.1007/978-1-4612-6290-9_6 +- zb: "0386.54001" name: Counterexamples in Topology --- -Asserted in the General Reference Chart for space #74 in -{{doi:10.1007/978-1-4612-6290-9_6}}. +The Euclidean balls contained in $\mathbb R^2\setminus\{0\}$ +and canonical base neighbourhoods of $0$ and $0^*$ are +interiors of their closures. diff --git a/spaces/S000066/properties/P000017.md b/spaces/S000066/properties/P000017.md index 27be847d66..ec95630095 100644 --- a/spaces/S000066/properties/P000017.md +++ b/spaces/S000066/properties/P000017.md @@ -3,9 +3,8 @@ space: S000066 property: P000017 value: true refs: -- doi: 10.1007/978-1-4612-6290-9_6 +- zb: "0386.54001" name: Counterexamples in Topology --- -Asserted in the General Reference Chart for space #74 in -{{doi:10.1007/978-1-4612-6290-9_6}}. +The sets $K_n:=\{p\in\mathbb R^2:\frac1n\leq\|p\|\leq n\}$ for $n\geq 1$ and $\{0,0^*\}$ form a compact cover of $X$. diff --git a/spaces/S000066/properties/P000022.md b/spaces/S000066/properties/P000022.md index 4b1bec65e7..ad79b515b5 100644 --- a/spaces/S000066/properties/P000022.md +++ b/spaces/S000066/properties/P000022.md @@ -2,10 +2,7 @@ space: S000066 property: P000022 value: false -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology --- -Asserted in the General Reference Chart for space #74 in -{{doi:10.1007/978-1-4612-6290-9_6}}. +The unbounded map $\mathbb R^2\ni (x,y)\mapsto x$ and $0^*\mapsto 0$ +can be readily verified to be continuous. diff --git a/spaces/S000066/properties/P000027.md b/spaces/S000066/properties/P000027.md deleted file mode 100644 index 2306d16ae0..0000000000 --- a/spaces/S000066/properties/P000027.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000066 -property: P000027 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -See item #2 for space #74 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000066/properties/P000038.md b/spaces/S000066/properties/P000038.md index 1e9daa6283..2c02a5c4b2 100644 --- a/spaces/S000066/properties/P000038.md +++ b/spaces/S000066/properties/P000038.md @@ -3,8 +3,8 @@ space: S000066 property: P000038 value: true refs: -- doi: 10.1007/978-1-4612-6290-9_6 +- zb: "0386.54001" name: Counterexamples in Topology --- -See item #3 for space #74 in {{doi:10.1007/978-1-4612-6290-9_6}}. +See item #3 for space #74 in {{zb:0386.54001}}. diff --git a/spaces/S000066/properties/P000042.md b/spaces/S000066/properties/P000042.md index 4a9549bc2f..e8d891254b 100644 --- a/spaces/S000066/properties/P000042.md +++ b/spaces/S000066/properties/P000042.md @@ -3,7 +3,7 @@ space: S000066 property: P000042 value: true refs: -- doi: 10.1007/978-1-4612-6290-9_6 +- zb: "0386.54001" name: Counterexamples in Topology --- @@ -12,4 +12,4 @@ and $V_{n}(0^{\ast})$ of $0^{\ast}$ is {P37}. And the same holds for small enough open balls around other points. Asserted in the General Reference Chart for space #74 in -{{doi:10.1007/978-1-4612-6290-9_6}}. +{{zb:0386.54001}}. diff --git a/spaces/S000066/properties/P000056.md b/spaces/S000066/properties/P000056.md deleted file mode 100644 index 28ab7f3244..0000000000 --- a/spaces/S000066/properties/P000056.md +++ /dev/null @@ -1,11 +0,0 @@ ---- -space: S000066 -property: P000056 -value: false -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -Asserted in the General Reference Chart for space #74 in -{{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000066/properties/P000082.md b/spaces/S000066/properties/P000082.md new file mode 100644 index 0000000000..b61f3323b3 --- /dev/null +++ b/spaces/S000066/properties/P000082.md @@ -0,0 +1,7 @@ +--- +space: S000066 +property: P000082 +value: true +--- + +Canonical base neighbourhoods have the topology induced from {S176}. diff --git a/spaces/S000066/properties/P000089.md b/spaces/S000066/properties/P000089.md new file mode 100644 index 0000000000..b75ad118d4 --- /dev/null +++ b/spaces/S000066/properties/P000089.md @@ -0,0 +1,9 @@ +--- +space: S000066 +property: P000089 +value: false +--- + +Consider the map $f:X\to X$ given by +$f((x,y))=(-x,-y)$ for $(x,y)\in\mathbb R^2\setminus\{0\}$ and $f(0)=0^*$, $f(0^*)=0$. +It has no fixed point and can be readily verified to be a homeomorphism. diff --git a/spaces/S000066/properties/P000110.md b/spaces/S000066/properties/P000110.md new file mode 100644 index 0000000000..0a54be29c2 --- /dev/null +++ b/spaces/S000066/properties/P000110.md @@ -0,0 +1,10 @@ +--- +space: S000066 +property: P000110 +value: true +--- + +For $x\in\mathbb R^2\setminus\{0\}$ we define $U_n(x):=B_e(x,2^{-n})\setminus\{0\}$. +Moreover, $U_n(0):=\{0\}\cup\{(x,y)\in\mathbb R^2: y>0,\ x^2+y^2<4^{-n}\}$ +and $U_n(0^*):=\{0^*\}\cup\{(x,y)\in\mathbb R^2: y<0,\ x^2+y^2<4^{-n}\}$. +The families $\mathscr V_n:=\{ U_n(x): x\in X\}$ can be readily verified to form a development. diff --git a/spaces/S000066/properties/P000129.md b/spaces/S000066/properties/P000129.md deleted file mode 100644 index 42c9bae4e2..0000000000 --- a/spaces/S000066/properties/P000129.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000066 -property: P000129 -value: false ---- - -The space is non-trivial by definition. diff --git a/spaces/S000066/properties/P000206.md b/spaces/S000066/properties/P000206.md new file mode 100644 index 0000000000..5e2423e694 --- /dev/null +++ b/spaces/S000066/properties/P000206.md @@ -0,0 +1,15 @@ +--- +space: S000066 +property: P000206 +value: true +--- + +The space contains precisely two points with neighbourhoods not homeomorphic to {S176}. + +Case 1. If player 1 picks $x_n\in\mathbb R^2\setminus\{0\}$, then +player 2 wins the game, since {S176|P206}. + +Case 2. If $x_n$ is one of the zero points, player 2 can +choose $V_n$ to exclude the other zero. Then either +$(x_m)_{m\geq n}$ is constant (and player 2 wins) or at some point we have Case 1. + \ No newline at end of file