Playful observations with deformation retracts and path connected-ness
I was going through a problem in Lee’s Topological Manifolds book1 (problem 7-12). It states that the infinite broom has a strong deformation retract to $(0, 0)$ but not for $(1, 0)$ (only a deformation retract). Below is some of my playful observations inspired by this problem. (Not restricted to the infinite broom space.) Inifinite Broom Note that the deformation retract to $(0, 0)$ itselfs imply the deformation retract to $(1, 0)$, since one can define the deformation retract to $(1, 0)$ by “first retract to $(0, 0)$, then push it to the point $(1, 0)$ along the line joining these two points”....