Is the letter X topologically equivalent to the letter Y?

Question

Is the letter X topologically equivalent to a Y? Can we squish one leg of the X to form a Y? This video seems to imply that we can't because X has four tails and Y only three, but I just want to double check if my understanding is correct. https://www.youtube.com/watch?v=QocXnf9bHo8 I was unsure because if I imagine X to be made of clay, I can squish one leg and turn it into a Y. Of course, I only know so little about topology so am not sure about the rules.

Answer 1

Deleting the centre point of the $X$ will split it into four connected components. There is no point on the $Y$ which will do this.

Answer 2

If by topologically equivalent you mean homeomorphic then the answer is no. There is no point in $Y$ the removal of which will split it into $4$ connected components, whereas there is one in $X$. This is a structural property that should be preserved via homeomorphisms.

However, if by topologically equivalent you mean that they have the same homotopy type, then yes, they are both contractible and have the homotopy type of a single point.