Literature references for the evident 'arc'-version of 'paths in a space compose'. Questions. Let 'arc in a topological space'$:=$'map $[0,1]\rightarrow X$ which is a homeomorphism onto its image'.


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*What do you consider usual citable references for the following statement?


Let $X$ be a Hausdorff space. Let $x_0,x_1,x_2\in X$ be mutually distinct. For both $i\in 2$ let $[0,1]\xrightarrow[]{A_i}X$ be an arc with $A_i(0)=x_i$ and $A_i(1)=x_{i+1}$. Then there exists an arc $[0,1]\xrightarrow[]{A_2}X$ with $A_2(0)=x_0$ and $A_2(1)=x_2$.


*What is the weakest separation axiom (known to you) on $X$ other the above-mentioned $\textsf{T}_2$ that makes the above conclusion true, and what are references for/discussions of the significance of, such more general 'arc-composition-theorem'? 


Remarks.


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*Without mutual distinctness of the points, $x_0=x_2=0\in\mathbb{R}$, $x_1=1$, $X=\mathbb{R}$ and $A_1$ and $A_2$ e.g. the linearly-parametrized arcs,  is a counterexample.

*Motivation for this question is working on problems about infinite planar (di)graphs. 

*related

*related
 A: I'm afraid I don't know of any references, but I hope the following discussion may nevertheless be helpful to you.
First, there is a very simple proof of this fact, so simple that I wouldn't be surprised if there is no standard reference.  You just truncate $A_0$ at the first point where it crosses $A_1$, and then concatenate the path $A_0$ up to that point with the remainder of $A_1$.  In detail: consider the set $$C=\{(s,t)\in [0,1]\times[0,1]:A_0(s)=A_1(t)\}.$$  Since $X$ is Hausdorff, $C$ is closed (it is the inverse image of the diagonal in $X\times X$ under the map $(A_0,A_1)$).  Thus $C$ is compact, and so its projection $$D=\{s\in[0,1]:A_0(s)=A_1(t)\text{ for some }t\in [0,1]\}$$ is also compact (and nonempty since $A_0(1)=A_1(0)$).  Let $s$ be the least element of $D$, and let $t\in [0,1]$ be such that $A_0(s)=A_1(t)$.  Now simply define $A_2$ to be the concatenation of the paths $A_0|_{[0,s]}$ and $A_1|_{[t,1]}$.  This path $A_2$ will be injective by minimality of $s$, and is thus an arc since any continuous injection from a compact space to a Hausdorff space is an embedding.  Since $x_0\neq x_2$, we have either $s>0$ or $t<1$, so the domain of the concatenation really is a nondegenerate interval and not just a single point.
Second, I think Hausdorffness is pretty much the only natural hypothesis to require here, and would be surprised if there is any interestingly weaker separation axiom that suffices.  I think the following example of a non-Hausdorff space where it fails is illustrative.  Let $X$ be $[0,1]$ with the endpoint $0$ "doubled": that is, $X=[0,1]\cup\{0'\}$ where neighborhoods of $0'$ are sets of the form $\{0'\}\cup(0,\epsilon)$.  There is an arc $A_0$ from $0$ to $1$ and an arc $A_1$ from $1$ to $0'$, namely $A_0(t)=t$ and $A_1(t)=1-t$ for $t<1$ and $A_1(1)=0'$.  But there is no arc from $0$ to $0'$, since any nonconstant path starting at $0$ must traverse all of $(0,\epsilon)$ for some $\epsilon>0$ and then in order to reach $0'$ it must pass back through the same interval.
The only separation axiom I can think of which is weaker than Hausdorff and suffices is that $X$ be weakly Hausdorff, meaning that the image of any continuous map from a compact Hausdorff space to $X$ is closed.  This suffices, since in that case the image of the concatenation of $A_0$ and $A_1$ is Hausdorff (the image of any map from a compact Hausdorff space to $X$ is Hausdorff; see Lemma 1.4(b) of http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf), and so you can just restrict your attention to that image.
