A topological groupoid structure on a pair $(X,A)$

Question

Assume that $X$ is a compact Hausdorff space and $A\subset X$ is a retract of $X$.

Is there a topological groupoid structure on the topological pair $(X,A)$ where, in the corresponding small category, $X$ and $A$ plays the role of morphisms and objects, respectively.

Edit 1: Is there a theory which investigate such type of problems for $A$ not necessarily a single point

In particular, is there a non single retract $A$ of $X=S^{3}$ or $X=S^{7}$ such that $(X,A)$ does not admit a topological groupoid structure.

In this question we do not require that the retracting map has any relation to the source and range maps

Edit 2: As another particular case, is there a groupid structure on $(G,G^{0})$ where $G=Gl(n,\mathbb{R})$ and $G^{0}=O(n)$? That is: are there range, source and composition maps which are continuous which makes $(G,G^{0})$ a groupoid.

Answer 1

Note: This answer had been sent to Ali in private email (the question had been closed while I was composing it). Now that the question has been reopened, the answer may be given here. (This has been edited, but the original letter is revision 1.)

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The answer to the question about $S^n$ and their hemispheres $H$ is yes, in a fairly easy way.

The inclusion $i: H \hookrightarrow S^n$ of the northern hemisphere has a retract $r: S^n \to H$ which linearly reflects points on the southern hemisphere across the equatorial hyperplane. Taking $C_1 = S^n$ and $C_0 = H$, and the domain and codomain maps $\partial_0, \partial_1: C_1 \rightrightarrows C_0$ both to be $r$, we may then define the object-to-identity map $u: C_0 \to C_1$ to be $i$, and we get a reflexive graph. Now we define composition to make this to a groupoid. the object $C_2 = C_1 \times_{C_0} C_1$ of composable pairs is just the kernel pair, i.e., the subspace of $C_1 \times C_1$ given by $\{(x, y): r(x) = r(y)\}$. Let $H'$ be the southern hemisphere and let $s: S^n \rightleftarrows H': j$ be the analogous retraction pair for the southern hemisphere, and define $m: C_2 \to C_1$ by $m(x, y) \mapsto i r(x)$ if $x = y$ and $(x, y) \mapsto j s(x)$ if $x \neq y$. It is not hard to check that this defines a continuous map.

The picture is that if you forget the topological structure and look at the underlying discrete groupoid, then there are no morphisms $a \to b$ if $a \neq b$, and the groupoid is a coproduct of groups $\bigsqcup_{a \in H} \hom(a, a)$ where $\hom(a, a)$ is the trivial group if $a$ lies on the equator, and $\hom(a, a)$ is the group $\mathbb{Z}/(2)$ if $a$ is off the equator. As a groupoid this is not connected (since there are no morphisms $a \to b$ between distinct $a, b$), although the total space $C_1$ is a connected space.