Counterexamples to Whitehead's Theorem for non-CW complexes

Question

Whitehead's theorem states that if X,Y are CW complexes and $f:X\to Y$ induces an isomorphism $f_* : \pi_n(X) \to \pi_n(Y)$ for all $n$, then $f$ is a homotopy equivalence. Are there simple counterexamples to this theorem in the case that $X$ and $Y$ are not CW complexes? I'm having trouble thinking about this, because I don't often think about spaces that aren't CW complexes, so I apologize if I've missed a simple counterexample.

Answer 1

Take $X$= one-point space, $Y$ = Warsaw circle and $f : X \to Y$ any map.

See How to show Warsaw circle is non-contractible?

Answer 2

Another classical example is the so called pseudocircle, the space on a four points set $X=\{a,b,c,d\}$ with open sets $\tau=\{\{a,b,c,d\},\{a,b,c\},\{a,b,d\},\{a,b\},\{a\},\{b\},\varnothing\}$. There is a continuous map $f\colon S^1\to X$ that induces an isomorphism on all homotopy groups, but the two spaces are obviously not homotopy equivalent.

This is a special case of a much more general counterexample, for every finite simplicial complex there is a finite topological space which is weakly homotopy equivalent but not homotopy equivalent to its geometric realization.

Answer 3

$X=$ one-point space, $Y= $ the long line, then any maps $X\to Y, Y\to X$ are weak equivalences but not homotopy equivalences (the long line is "too long" to be shrunk to a point by a homotopy)

Another example (which is an example for a different reason) is $\mathbb Q$ with the discrete topology, which I'll denote $\mathbb Q^{dis}$ and $\mathbb Q$ with the usual topology : $id : \mathbb Q^{dis}\to \mathbb Q$ is continuous and is a weak equivalence, but it's not a homotopy equivalence (any map $\mathbb Q\to\mathbb Q^{dis}$ which induces an isomorphism on $\pi_0$ must be the identity, and it's not continuous)

More generally, for any space $X$ you can find (explicitly) a CW-complex $Y$ and a weak equivalence $Y\to X$. So if you know for some reason that $X$ does not have the (strong) homotopy type of a CW complex, then you can use this as an example. This will work for, e.g. any non- (semi-locally simply connected) space, such as the earrings space.