Question: Let $A$ be a collection of 3 closed line segments (of finite length) in the plane. Construct as many non-homeomorphic subspaces of the plane which are the union of these 3 line segments under the condition that when the lines intersect, the intersection is a singleton.
I've started drawing lots of circles and lines intersecting them and I'm worried I'm barking up the wrong tree. Could you help me understand what this question is asking and help me find some of the more difficult subspaces?
Edit: Changed 'union' 'intersection'.
Any required subspace $X$ is an image of a plane embedding of the graph $G_X$ whose set of vertices is the set of endpoints and the intersection points of the segments of the collection. We can easily list all such graphs $G_X$ by the induction with respest to the number of $n$ segments. This gives the following lists of the possible required subspaces.
$n=1$:
I
$n=2$:
II, I, X, Y
$n=3$:
It is easy to check (mainly by looking of sets of connected components created after removing crossroad points) that all listed subspaces are mutually non-homeomorphic.