Remark 2

Let X and Y be topological spaces and let $f: X\rightarrow Y$ be a continuous mapping. The canonical mapping $\alpha '_f: X\times \mathbf{I}\rightarrow$ Cone$(f)$ is constant on $X\times \{0\}$ and therefore defines a continuous mapping $\gamma_f: C(X)\rightarrow$ Cone$(f)$. The restriction of the mapping $\gamma_f$ to $C'(X)$ is injective and strict, and defines by passing to the subspaces a homeomorphism of the open cone $C'(X)$ onto the complement of Y in Cone$(f)$.

Let us identify the base of the cone C(X) with the space X and let $u$ denote the continuous mapping of $Y\cup_fC(X)$ into Cone$(f)$ deduced from the mappings $\beta '_f: Y\rightarrow$ Cone$(f)$ and $\gamma_f: C(X)\rightarrow$ Cone$(f)$. Conversely, let $v:$ Cone$(f)\rightarrow Y\cup_fC(X)$ be the unique continuous mapping such that $v\circ \alpha '_f$ is the canonical mapping of $X\times \mathbf{I}$ onto C(X) and $v\circ \beta '_f$ is the canonical mapping of Y onto $Y\cup_fC(X)$. The mappings $u$ and $v$ are inverse homeomorphisms of one another.