Properties
All cones are path-connected since every point can be connected to the vertex point. Furthermore, every cone is contractible to the vertex point by the homotopy
- ht(x,s) = (x, (1−t)s).
The cone is used in algebraic topology precisely because it embeds a space as a subspace of a contractible space.
When X is compact and Hausdorff (essentially, when X can be embedded in Euclidean space), then the cone CX can be visualized as the collection of lines joining every point of X to a single point. However, this picture fails when X is not compact or not Hausdorff, as generally the quotient topology on CX will be finer than the set of lines joining X to a point.
Read more about this topic: Cone (topology)
Famous quotes containing the word properties:
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—John Locke (1632–1704)
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—Ralph Waldo Emerson (1803–1882)