Meaning of Quotient in this context

Question

I was seeing the following problem a couple days ago: Let $R \subset \mathbb{R}^2$ denote the unit square $R = [0,1] \times [0,1]$. If $F \subset R$ is finite, is $R \backslash F$ connected?

I understand $R\backslash F$ in the case where F is an equivalence relation but in this it does not seem like it. Can someone explain this to me in a intuitive way?

Answer 1

$R\setminus F$ is just a square without finite number of points. It seems to be connected, because each two points of $R\setminus F$ can be easily connected by a broken line consisting of two segments.

Answer 2

No; the image will always be connected (in the quotient topology), because a quotient map is , by construction/definition continuous, and the continuous image of a connected space (here the unit square) is always connected.