Cone (topology)
Examples
Examples
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Here we often use a [geometric cone](/wiki/Geometric_cone "Geometric cone") (C X where X is a non\-empty [compact](/wiki/Compact_space "Compact space") subspace of [Euclidean space](/wiki/Euclidean_space "Euclidean space")). The considered spaces are compact, so we get the same result up to homeomorphism.

* The cone over a point *p* of the [real line](/wiki/Real_line "Real line") is a line\-segment in \\mathbb{R}^2, \\{p\\} \\times \[0,1].
* The cone over two points {0, 1} is a "V" shape with endpoints at {0} and {1}.
* The cone over a [closed interval](/wiki/Closed_interval "Closed interval") *I* of the real line is a filled\-in [triangle](/wiki/Triangle "Triangle") (with one of the edges being *I*), otherwise known as a 2\-simplex (see the final example).
* The cone over a [polygon](/wiki/Polygon "Polygon") *P* is a pyramid with base *P*.
* The cone over a [disk](/wiki/Disk_%28mathematics%29 "Disk (mathematics)") is the solid [cone](/wiki/Cone_%28geometry%29 "Cone (geometry)") of classical geometry (hence the concept's name).
* The cone over a [circle](/wiki/Circle "Circle") given by

\\{(x,y,z) \\in \\R^3 \\mid x^2 \+ y^2 \= 1 \\mbox{ and } z\=0\\}
is the curved surface of the solid cone:
\\{(x,y,z) \\in \\R^3 \\mid x^2 \+ y^2 \= (z\-1\)^2 \\mbox{ and } 0\\leq z\\leq 1\\}.
This in turn is homeomorphic to the closed [disc](/wiki/Disc_%28mathematics%29 "Disc (mathematics)").
More general examples:*, Section 4\.3*
* The cone over an [*n*\-sphere](/wiki/N-sphere "N-sphere") is homeomorphic to the closed (*n* \+ 1\)\-[ball](/wiki/Ball_%28mathematics%29 "Ball (mathematics)").
* The cone over an [*n*\-ball](/wiki/N-ball "N-ball") is also homeomorphic to the closed (*n* \+ 1\)\-[ball](/wiki/Ball_%28mathematics%29 "Ball (mathematics)").
* The cone over an *n*\-[simplex](/wiki/Simplex "Simplex") is an (*n* \+ 1\)\-simplex.

Can you give me some examples of compact subspaces in Euclidean space?