Htykuut

Does there exist a topological space $X$ and a closed subspace $A$ such that each of $X$ and $A$ is isomorphic to the topological space of some simplicial complex, but such that there does not exist a pair $(S,T)$ consisting of a simplicial complex $S$ and a subcomplex $T$ with $(X,A) cong (|S|,|T|)$?

This is (essentially) problem A2 from Chapter 3 of Spanier's book Algebraic topology. At first I suspected that the intended example was the comb space $A$ inside the unit square $X$, but it seems that $A$ is not polyhedral (if it were, then being compact, it would have to be the space of a finite simplicial complex). It doesn't seem to me that there is an example that has already appeared in Spanier's book, so I am somewhat at a loss as to the intended answer.

One example is the Alexander horned sphere. This is a pathological embedding of $S^2$ into $mathbb{R}^3$, which in particular has the property that the unbounded component of its complement has non-finitely generated fundamental group. If you take $A$ to be the horned sphere and $X$ to be a closed ball containing it, then $A$ and $X$ are both finite simplicial complexes, but one of the components of $Xsetminus A$ has non-finitely generated fundamental group. If $X$ had a simplicial structure such that $A$ was a subcomplex, then this simplicial structure could only have finitely many cells since $X$ is compact, and then it is easy to see that $pi_1(Xsetminus A)$ would be finitely generated for any basepoint.