Open Thickening of Closed Balls - Homotopy Equivalent? Let us assume we have a finite union of closed ball with radius $\alpha$ around points $x$ in a Euclidean space
$$M:= \bigcup_{i \in \{1,...,q\}}B_\alpha(x_i)$$
Can I &quot;thicken&quot; them to open balls $B^\circ_{\alpha +\epsilon}(x_i)$ such that
$$M \text{   is homotopy equivalent to   } \bigcup_{i \in \{1,...,q\}}B^\circ_{\alpha+ \epsilon}(x_i)$$
It should work, but could you tell me the reason why? Thanks a lot!!
I think the concept underlying your question is that of a deformation retract. (See here: https://en.wikipedia.org/wiki/Retraction_(topology)).
The idea is that you can deformation retract on your union to get the same thing. Deformation retracts are special kinds of homotopy equivalence.
I think the concept underlying your question is that of a deformation retract. (See here: https://en.wikipedia.org/wiki/Retraction_(topology)).
The idea is that you can deformation retract on your union to get the same thing. Deformation retracts are special kinds of homotopy equivalence.