TITLE: Existence of weak homotopy equivalence not a symmetric relation
QUESTION [9 upvotes]: 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.

REPLY [8 votes]: 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.