# Explanation of Lemma 2.34 in Hatcher's ''Algebraic Topology"?

In this proof of the Lemma 2.34 of Algebraic Topology I don't understand 2 things.

$(1)$ Hatcher defines $Y_i:=T\cup (X\times[i,\infty])$, then I think that it's obvius the fact that $Y_i$ deformation retracts onto $Y_{i+1}$, since $[i,\infty]$ deformation retracts onto $[i+1,\infty]$.

$(2)$ With the retractions of $Y_i$ in $Y_{i+1}$ how I can obtain a deformation retract of $X\times [0,\infty)$ onto $T$?

• I've edited your title a bit, since it doesn't seem like you're actually claiming that something is wrong with Hatcher's proof; you're asking for clarification on two points. Feel free to rollback the edit if you see fit. Feb 5 '17 at 21:25
• Thanks :) Now...Can you help me with the problem? I'm desperate! xD Feb 5 '17 at 21:27

Concerning (1), we cannot just expect a deformation retraction of $$X\times \lbrack i,\infty )$$ to $$X\times \lbrack i+1,\infty )$$ to extend to a deformation retraction of the larger space $$T\cup X\times\lbrack i,\infty )$$. For instance, $$S^1$$ is the union of its northern and southern (closed) hemispheres. Both hemispheres deformation retract to a point, but there is no deformation retraction of $$T$$ which, e.g., is the identity on the southern hemisphere while deforming the northern hemisphere to a point.
As for (2), the point is to compress each $$Y_i$$ in increasingly short time intervals. The resulting homotopy is well-defined and continuous, essentially due to the "pasting lemma" from elementary point-set topology; the intervals $$\lbrack 1 - 1/2^i, 1 - 1/2^{i+1}\rbrack$$ form a closed cover of the unit interval and the partial homotopies agree on the intersections $$\lbrace 1/2^i\rbrace$$ by construction. This allows us to eventually compress every $$Y_i$$ (this is no weirder than Achilles catching up to the turtle).