Let $X=C[0,1]$ be the normed vector space with maximum norm. Give it the norm-topology. Let $Y=X-\{0\}$. Find $H_n(Y)$.
Intuitively, I guess the answer is 0 for all $n$, but I have no idea.
2026-03-29 03:36:08.1774755368
Homology Group of Infinite-dimensional Vector Space without a point
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Hint: Consider $H_t(x)={{tx}\over{\|x\|}}+(1-t)x$ It it a deformation retract between $S^{\infty}$ and $E-\{0\}$. $S^{\infty}$ is contractible.