Let us call a nonempty topological space a *topological tree* if it is Hausdorff and for two distinct points there is a continuous injective path connecting the points, which is unique up to reparametrisation.

This should be equivalent to the following definition:
The space $X$ is a *topological tree* if it is Hausdorff, pathwise connected and whenever $\gamma_1,\gamma_2:[0,1]\to X$ are topological embeddings with $\gamma_1(0)=\gamma_2(0)$ and $\gamma_1(1)=\gamma_2(1)$, it is true that $\gamma_1([0,1])=\gamma_2([0,1])$.

I am not reqiring the space to be paracompact nor to have the homotopy type of a CW-complex or anything like that, just a Hausdorff space.

Examples of such *topological trees* are $\mathbb R$, a hedgehog space (https://en.wikipedia.org/wiki/Hedgehog_space), or the Long Line (or the Long Ray).

What one could expect that such a space is always contractible as it is called a *tree* and trees should be contractible but the example of the Long Line shows that this is not the case.

However, it **should** be true that such a space is *weakly contractible*, i.e. every homotopy group is trivial. As $\pi_0$ is trivially trivial, I tried to show that for the fundamental group $\pi_1$ but was not able to do so.

Maybe this is a hard question like the one why two different points in a pathwise connected Hausdorff space can always be connected by an injective arc. I know where to find a proof for that fact, but is it frustrating that there seems to be not *simple* proof for that fact that seems obvious but is not.

By the way: I do not know if that defintion of a topological tree is already in use under a different name. I just took the notion of an $\mathbb R$-tree from metric geometry and removed the metric geometry out of the definition.

Another question would be if a *topological tree* is always 1-dimensional, using different notions of topological dimensions, but that should be a different question for a different day.

pathwise connectedness, notpairwise connectedness. $\endgroup$real trees, see en.wikipedia.org/wiki/Real_tree $\endgroup$1more comment