Hatcher, Algebraic Topology, Chapter 0
6. (a) Let be the subspace of consisting of the horizontal segment together with the vertical segments for a rational number in . Show that deformation retracts to any point in the segment , but not to any other point (see the preceding problem).
(b) Let be the subspace of that is the union of an infinite number of copies of arranged as in the figure below. Show that is contractible but does not deformation retract onto any point.
(c) Let be the zigzag subspace of homeomorphic to indicated by the heavier line. Show that there is a deformation retraction in the weak sense (see Exercise 4) of onto , but no true deformation retraction.
Proof. (a) Let be the subspace of shown on the far left of Figure 1 consisting of the horizontal segment together with the vertical segments for a rational number in . First, note that the segment deformation retracts onto any point by way of the straight-line homotopy which fixes for all and which linearly “squeezes” the intervals and towards their right and left endpoints, respectively. Algebraically, then, is the family of maps for which
Next, note that the space deformation retracts onto the space in a fashion similar to the one above, namely by way of the family of maps which, for each rational , maps the segment to the point continuously with respect to as ranges through . Denoting this family of maps , it follows that the space deformation retracts to any point by way of the composition , where
Finally, the goal is to show that fails to deformation retract onto any point not on the segment . For a contradiction, suppose that there does exist such an for which a deformation retraction exists. By the result of Exercise 0.5, it follows that every neighborhood of in contains a neighborhood of for which the inclusion is null-homotopic. Evidently this is bad; how come?2
Note that is path-connected1 but that clearly isn’t path-connected. In particular, suppose is a neighborhood of in . Then can be thought of as an open ball in , disjoint from , and intersecting a countably infinite number of line segments, all of which are disjoint. Thus, fails to be path-connected. For the same reason, any neighborhood containing must also be path-disconnected, whereby it follows that cannot contract to a point. Hence, cannot be homotopic to the constant map, contradicting the result of 0.5.
(b) Let be the subspace of shown in the rightmost part of Figure 1 consisting of an infinite number of copies of arranged as above. The fact that fails to deformation retract onto any point is an almost-identical application of Exercise 0.5 to the one mentioned in (a). In particular, given a point , there exists neighborhood of in that is necessarily path-disconnected. This claim follows immediately from (a) if fails to lie on the zigzag subspace (see part (c)). In the case that , this can be heuristically described in cases as follows.
Suppose fails to be one of the “vertex points”. Any neighborhood in intersects two “fundamentally different” countably infinite disjoint collections of line segments of , one collection which is directed “bottom-left-to-top-right” – call it – and one which is directed “top-left-to-bottom-right” – call it . In this case, any point on a segment in cannot be connected to any point in because of the fact that intersects only one of , .
The argument for a “vertex point” of is similar with the exception that an open neighborhood of now intersects three fundamentally different collections of line segments, two of which – say and – are directed in a parallel fashion (either “bottom-left-to-top-right” or “top-left-to-bottom-right”) with the third, , directed in the other fashion. In this case, any point in , , can be connected to any point in by a path due to the fact that intersects and one of , . On the other hand, intersections precisely one of or , meaning that no path in exists connecting points in to .
Now, it suffices to show that is contractible. Note, first, that is clearly contractible. Next, note that the space can be retracted (not deformation retracted) onto by way of a modified version of the family given in part (a). More precisely, the map is a retraction, where is the constant map on and where maps any “straw” onto the point , where the subscript here denotes the elements in the copy of .3 Therefore, it’s been shown that retracts onto the zigzag which is a contractible subspace, thereby proving that itself is contractible.
(c) To begin, note that there is clearly no (strong) deformation retract of onto , else the composition with the deformation retraction of to a point as noted in (b) would yield a deformation retraction of to a point. This contradicts the result of (b). As such, it’s enough to show that the “contraction” of onto the zigzag subspace is a (weak) deformation retraction. To that end, let be the family which fixes the space and takes each “leaf” and shrinks its length by a factor of , i.e.
for all (see ).
Continuity here is immediate, and because the three conditions outlined in Exercise 0.4 for weak deformation retractions are clearly met, the result follows.
1. The path-connectedness of is easily shown by an analysis of cases for arbitrary points . If , the result is immediate. If one of the pair is in and the other isn’t (WLOG, and ), then the path works. Finally, if and , then the path works as well.
2. Ideas for the final part of this proof were borrowed from the document found here: http://www.math.uchicago.edu/~may/263/ps1.pdf.