Cone space

This article describes a construct that involves some variant of taking a product of a topological space with the unit interval and then making some identifications, typically at the endpoints, based on some specific maps.
View more such constructs

Definition

Given a topological space $X$, the cone space of $X$, denoted as $CX$, is defined as the quotient of $X\times I$ (where $I$ is the closed unit interval $[0,1]$ and we use the product topology) by the equivalence relation:

$(x_{1},0)\sim (x_{2},0)\forall x_{1},x_{2}\in X$

Here, $I$ refers to the closed unit interval $[0,1]$.

Refer:

• Cone space functor to see the properties of the cone space functor
• Cone-realizable space to see the property of a topological space being realizable as the cone space over some space