We have the following set up:
$X_0 \subset X_{\pm} \subset X$. Also interiors of $X_\pm$ cover $X$. Now let $Z$ be the double mapping cylinder of the maps $X_- \leftarrow X_0 \rightarrow X_+$. Define $N(X_-,X_+)=X_-\times 0 \cup X_0 \times I \cup X_+ \times 1$ with the subspace topology inherited from $X \times I$. Now there is a canonical bijective map $\alpha : Z \to N$ over $X$. The author (on page 86 in Algebraic Topology book by Tammy Tom Dieck) claims that this map is an h-equivalence.
At first glance I thought that $\alpha$ is a homeomorphism, but now my intuition says that it need not be. Could someone please help me in constructing a counter example ?
- Basis for my intuition : Let $Y=\mathbb{R}$, $Y_1=\mathbb{R_+} \cup 0$ and $Y_2=(Y-Y_1) \cup [5, \infty)$. Let $Y$ have the usual topology and let $Y_1,Y_2$ have the subspace topology. Then the topology on $Y$ induced by constructing the quotient space of $Y_1$ and $Y_2$ (by identifying pairs of point in intersection $Y_1 \cap Y_2$) is not the same as the usual topology on $Y$.
However I am unable to construct a similar example for the required case. Thanks !