Let $(X,A)$ be a "good" CW pair. Let $*\in A \subset X$ be the base point of $ X$ and $CA$ the cone on $A$ i want to show that $(X\cup CA)/CA$ is homeomorphic to $X/A$. I can see it geometrically but i want to prove it. Consider the composite $f:X\cup CA \to X \to X/A $ where the first map sends $x$ to $x$ and $(a,t)\in CA$ to $a$ and the second map is the quotient map. $f$ is a surjective map sending $CA$ to the base point of $X/A$ which is the class of $*$ identified with $A$ so $f$ factorizes through $(X\cup CA)/CA$ and the induced map
${\tilde f}:(X\cup CA)/CA\to X/A$ is a homeomorphism. Is this correct?