But don't assume knowledge of any topological dimension theory. Here is a specific approach (an open problem):
Does there exist a separable metric space $X$ such that the following two conditions hold:
- $|X|\ge 2$;
- for every closed $A\subseteq X$ such that $X\setminus A$ is disconnected, there exists $Y\subseteq A$ homeomorphic to $X$;
(I formulated this question around 1960, perhaps without ever publishing it; I didn't work on it--too bad, since I like this question :-) )
BTW, you're welcome to use any means (including the dimension theory) to construct $X$. If you did it without using dimension theory or fpp for $I^n$ then you'd obtain a new proof of $\dim I^n\ge n$ and of fpp for $I^n$, for every $n$ (it's well known and easy that if this fpp holds for every natural $n$, then for arbitrary cardinal $n$ too).
TERMINOLOGY: fpp = Fixed Point Property.