cone over space

Definition

Given a topological space $X$, the cone over $X$ is the topological space

$$\text{cone}(X):=[0,\infty )\times X/(\{0\}\times X).$$

Notes

This is the quotient topology given that we map all the points on the “bottom” of the interval to the same point.

Another way to think about this is that for the space $X$, when we product with an interval, we get a “cylinder” of the space $X$. The cone takes and “crushed” one side of the cylinder all down to a single point.