In the book of Buskes and van Rooij on Topological Spaces, there is an interesting introductory example on how the notion of topology arose from the distinction between several geometrical forms. Rigorously speaking, two forms are topologically distinguishable if they are not homeomorphic.
Consider the topology of the two letters: $T$ and $C$, on the plane. They are not diffeomorphic (at the intersection point of the letter $T$). The above textbook seems to imply that they are not homeomorphic.
The two curves are compact, simply connected. I tried to construct an explicit homeomorphism, but no advance for now.
My question is, are these two curves homeomorphic? If yes, what might be a homeomorphism? If no, why?
They are not homeomorphic. You can remove a point from $T$ that cuts it into three connected parts. You can't with $C$. That prevents a homeomorphism.
The $T$ isn't a "curve" - you can't parameterize it with a line segment.