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I am having trouble coming up with an example of spaces where there exists a weak homotopy equivalence in one direction but not the other. Any hints or references are greatly appreciated!

Note: This is an instance of stagnating autodidactic studying, hence no home-work tag.

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The circle $S^1$ is weakly equivalent to the so-called pseudocircle $\mathbb{S}$ (see wikipedia), and the weak equivalence goes $S^1 \to \mathbb{S}$. Any map $\mathbb{S}\to S^1$ induces the trivial map on $\pi_1$.

There are many more examples:

As shown by McCord (Singular homology groups and homotopy groups of finite topological spaces), any finite simplicial complex is weakly equivalent to a finite topological space.

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This is a cool answer! –  Rasmus Nov 8 '11 at 17:26
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