In another MathOverflow post I asked: In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?

Note that once consequence of this is that for spaces in which this holds you only need to consider simple loops when proving that a space is not simply connected. This also suggests that looking at path-connected spaces only would still be interesting.

Some very elegant and interesting counterexamples were given, both Hausdorff (for a space embedded in $\mathbb{R}^3$) and non-Hausdorff.

**I propose a follow up question which is to determine the spaces with the property that if there exists a loop that cannot be contracted to a point there exists a simple loop that cannot be contracted also?.** (Or equivalent to classify the counterexamples in some way)

For example what conditions can be place on a space embedded in $\mathbb{R}^n$ to force the condition to hold?

Update: Given that this is quite a broad question I have asked another question specifically about the $\mathbb{R}^2$ case: In a subset of $\mathbb{R}^2$ which is not simply connected does there exist a simple-loop that contracts to a point?