As far as I know, a quotient space of a topological space must be defined wrt an equivalence relation on the topological space.
But then I wonder what the equivalence relation is in the following example from Wikipedia:
In topology, especially algebraic topology, the cone $CX$ of a topological space $X$ is the quotient space: $ CX = (X \times I)/(X \times \{0\})\, $ of the product of $X$ with the unit interval $I = [0, 1]$. Intuitively we make $X$ into a cylinder and collapse one end of the cylinder to a point.
Is a subset of a topological space able to induce an equivalence relation on the topological space, and to induce a quotient space? I know it is true for a subspace of a vector space.
Thanks and regards!