I am struggling to understand basic topological questions about CW complexes. I understand the idea for showing that they are locally contractible, but I don't accept the proof yet, several details are missing in my mind.
((( Basic idea : You take a point and an open neighborhood, and construct, one $n$-skeleton at a time, a growing family of open contractible neighborhoods in said $n$-skeleton : they are obtained by thickening the 'traces' of the previous, lower dimensional neighborhood in the newly attached cells, staying small enough to still fit inside the original neighborhood.)))
My problem arises when I want to show that certain homotopies between maps defined on (open) subset of CW complexes are continuous.
I know one way of how to show that homotopies of maps defined on quotient spaces are continuous. If $X$ is a topological space and $\mathcal{R}$ an equivalence relation on $X$, then the canonical continuous bijection $$\frac{X\times I}{\mathcal{R}\times 1}\rightarrow X/\mathcal{R}\times I,~[(x,t)]\mapsto([x],t)$$ is a homeomorphism ($I=[0,1]$, and $\mathcal{R}\times 1$ is the equivalence relation on $X\times I$ such that $(x,t)\sim (x',t')$ iff $x\mathcal{R}x'$ and $t=t'$).
I'll phrase my $~~~~~~\mathrm{QUESTIONS}~~~~~~$ in the framework of CW complexes, but they are actually about subspaces of inductive limits and products with other spaces.
Suppose $(X,(X_n)_{n\in\mathbb{N}})$ is a CW complex, and I have a subset $Y\subset X$, and I define for all $n\in\mathbb{N},~Y_n=Y\cap X_n$, then I can define on $Y$ two topologies : the subspace topology (as a subspace of $X$) and the limit topology associated with the inclusions $$Y_0 \hookrightarrow Y_1 \hookrightarrow Y_2 \hookrightarrow \cdots \hookrightarrow Y_n \hookrightarrow Y_{n+1} \hookrightarrow \cdots $$ it follows from definitions that the canonical map $$Y_{\mathrm{ind.lim.}}\hookrightarrow Y_{\mathrm{subspace}}$$ is a continuous bijection.
Similarly, i have at least $2$ natural topologies on $Y\times I$ : the inductive limit topology $(Y\times I)_{\mathrm{lim.ind.}}$ of $$Y_0 \times I \hookrightarrow Y_1\times I \hookrightarrow Y_2\times I \hookrightarrow \cdots \hookrightarrow Y_n\times I \hookrightarrow Y_{n+1}\times I \hookrightarrow \cdots $$ and the product topology $Y_{\mathrm{ind.lim.}}\times I$ (there is also the product topology $Y_{\mathrm{subspace}}\times I$). As before, there is a canonical continuous bijection $$(Y\times I)_{\mathrm{lim.ind.}}\hookrightarrow Y_{\mathrm{ind.lim.}}\times I~~(\hookrightarrow Y_{\mathrm{subspace}}\times I)$$
It seems to me some proofs I've come across where one constructs homotopies between maps defined on a CW complex tacitly assume some of these maps to be homeomorphisms. So, when are they homeomorphisms?
Also, if anybody can direct me to a good reference on topologies of inductive limits etc I'd be grateful!
