Topology of two letters: showing that two letters are homeomorphic (or not)

Question

Hoe can i check if the letters T, K homemorphic or not? Same about the letters E, W? I guess that it can be shown by using connectedness and connected components but I have no idea how it can be applied. Can you help me in that?

Answer 1

If $f \colon T \to K$ is a homeomorphism and $x \in T$, then $f \colon T \setminus \{x\} \to K \setminus \{f(x)\}$ is also a homeomorphism. Now, using your idea of connectedness and connected components, apply this to a very particular element of $T$ and count the number of connected components in $T \setminus \{x\}$ and in $K \setminus \{f(x)\}$. They should be equal, but for a very particular choice of $x$ they won't be.

A similar trick can be applied to $E$ and $W$.