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hbghlyj 发表于23:59
I have a basic topology question. Let X be the set of natural numbers {0, 1, 2, ...} and let Y be the set {0} ∪ {1, 1/2, 1/3, ...}, both with the subspace topology from the real line. Are they homotopy equivalent spaces?

Thorgott 发表于23:59
@hbghlyj what do you think? how many maps from the latter to the former do you know?

hbghlyj 发表于00:17
Suppose $f:Y\to X$ is a continuous map. Since $f^{-1}(\{f(0)\})$ is open in $Y$ and contains 0, it must be of the form $\{0\}\cup\{\frac1n:n\ge k\}$ for some $k$.

Thorgott 发表于00:18
so what does that mean in terms of the connected components?

hbghlyj 发表于01:07
$f:Y\to X$ and $g:X\to Y$ induce $f _ *:\pi _ 0(Y) \to \pi _ 0(X)$ and $g _*:\pi_0(X)\to\pi _ 0(Y)$ and suppose $g\circ f\simeq \text{id}_Y$ then $g_* \circ f _* =\text{id}_*$ but $f_*$ is not injective, contradiction.
So $X$ and $Y$ are not homotopy equivalent.

Thorgott 发表于13:32
@hbghlyj yup, that's correct :)