S225: Closed upper half-plane

spaces/S000225/README.md

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+---
+uid: S000225
+name: Closed upper half-plane $\mathbb R^2_+$
+---
+
+The subspace $\mathbb R^2_+=\{(x,y)\in\mathbb R^2 : y\ge 0\}$ of {S176}.
+
+This space is homemorphic to the product of {S25} and {S210}.

spaces/S000225/properties/P000016.md

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+---
+space: S000225
+property: P000016
+value: false
+---
+
+$X$ is not a bounded subset of $\mathbb R^2$.

spaces/S000225/properties/P000089.md

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+---
+space: S000225
+property: P000089
+value: false
+---
+
+The map $(x,y)\mapsto (x+1,y)$ has no fixed point.

spaces/S000225/properties/P000120.md

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+---
+space: S000225
+property: P000120
+value: false
+---
+
+$X$ contains an open subspace homeomorphic to {S176}
+and {S176|P120}.

spaces/S000225/properties/P000122.md

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+---
+space: S000225
+property: P000122
+value: false
+---
+
+The point $(0,0)\in X$ does not have a neighborhood homeomorphic to some $\mathbb R^n$.

spaces/S000225/properties/P000184.md

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+---
+space: S000225
+property: P000184
+value: true
+---
+
+By definition.

spaces/S000225/properties/P000199.md

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+---
+space: S000225
+property: P000199
+value: true
+---
+
+Since $X$ is a convex subset of $\mathbb R^2$,
+it can be deformation retracted to a point using a straight-line homotopy.

spaces/S000225/properties/P000204.md

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+---
+space: S000225
+property: P000204
+value: false
+refs:
+  - mathse: 214317
+    name: $(X\times Y)\setminus (A\times B)$ is connected for $A,B$ proper subsets of connected $X,Y$
+---
+
+$X$ is the product of two connected spaces, each with at least two points.
+Such a product minus a point is always connected
+(see {{mathse:214317}}).

spaces/S000225/properties/P000236.md

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+---
+space: S000225
+property: P000236
+value: true
+---
+
+$X$ satisfies {P236} with $n=2$.