Vtyjkyuk

In Tom Dieck's algebraic topology, pg 92, he defines and states

The (pointed) cone $CX$ over $X$ is $X times I/ X times 0 cup * times I$. The inclusion $i_1^X: X rightarrow CX , x mapsto (x,1)$ is an embedding.

Why is this true? In fact if we take $C$ to be closed in $X$, let $q:X times I rightarrow CX$ denote the quotient map, then $$q^-1[i_1[C]]= C times 1 cup * times I$$ if $* in C$. $i_1$ is not necessarily a closed map to its image, hence not an embedding.

Note: no where does Tom Dieck states he is working in category of CGWH spaces. Otherwise this statement would be true as $*$ is closed in $X$.

Let $X' = q(X times 1) subset CX$. We have to show that $j : X to X', j(x) = q(x,1)$, is a homeomorphism. It is obviously a continuous bijection. Let $A subset X$ be closed. We want to show that $j(A)$ is closed in $X'$. This means that there exists $B subset CX$ such that $B cap X' = j(A) = q(A times 1)$ and $q^-1(B)$ is closed in $X times I$.

Case 1: $ast notin A$.

Take $B = q(A times 1)$. Then $q^-1(B) = A times 1$ is closed in $X times I$ and $B cap X' = q(A times 1) cap q(X times 1) = q(A times 1)$.

Case 2: $ast in A$.

Take $B = q(A times I)$. Then $q^-1(B) = A times I cup X times 0$ is closed in $X times I$ and $B cap X' = q(A times I) cap q(X times 1) = q(A times 1)$. Note that $q(A times I) = q(A times 0, 1 cup ast times I) cup q((A backslash ast) times (0,1)) = q(A times 1) cup q((A backslash ast) times (0,1))$.

BUT: $i_1^X : X to CX$ is in general not a closed embedding. It is one if and only if $q^-1(X') = X times 1 cup ast times I$ is closed in $X times I$. The latter is equivalent to $ast$ being closed in $X$.