I'm doing an exercise that is far more difficult than the others on my notes. I've been thinking about it but I don't know if my approach is the correct one.
Let $X$ and $Y$ be topological spaces. Let $f:X\to Y$ be a continuous function. We define the cylinder $C_f=((X\times[ 0;1])\sqcup Y)/\sim$, where $\sim$ is the relation $(x,0)\sim f(x)$.
i) Prove that $X$ and $Y$ are included in $Cf$.
ii) Prove that $X$ and $Y$ are closed subspaces.
iii) Prove that $r:C_f\to Y$ where $r([(x,t)])=f(x)$ and $r([y])=y$ is a retraction and it's continuous.
My loose thoughts on the tree parts:
i) Here I took the inclusion $i: X \to C_f$ where $i(x)=[(x,0)]$. So obviously $X$ it's included in $Cf$. Same with $Y$, we take the map $i: Y \to C_f$ where $j(y)=([y])$. Is that correct?
ii) $X$\ $C_f=\emptyset$, obviously is open (and closed) in the topological space $C_f$. Y is more difficult, because we don't know if $f$ is exhaustive. $Y$ \ $C_f=f(X)$ that is open (because $f$ is continuous). So both $X$ and $Y$ are closed.
iii) Here we have to see that $r\circ j=id_y$, where $j$ is the inclusion $j:Y \to C_f$, but I'm stuck.
Am I right? I see everything like a little bit "obvious", so almost sure I'm doing mistakes in every part.
Thanks for your time.
i) The relation $(x,0) \sim f(x)$ says that the slice $X \times \{0\}$ is a copy of $f(X) = Y$. However, every other slice (choice of second coordinate from $(0,1]$ is a copy of $X$, so, for example, $\{(x,1) \in C_f : x \in X\}$ is a copy of $X$. There is still a little work to show that the pointwise bijections "is a copy of" are homeomorphisms.
ii) It is far easier to notice that the projection $X \times [0,1] \rightarrow [0,1]: (x,y) \mapsto y$ is continuous. The two slices we mentioned above are closed if they are the (full) preimages of closed subspaces of $[0,1]$. (And you know a lot more about closed subspaces of $[0,1]$ in the subspace topology of the standard topology on $\mathbb{R}$, so this should be easy.)
iii) I agree with Tyrone's comment. The work to show here is that you actually have continuous maps.