I have this problem
"Let $X$ be the space obtained from the torus $S^1\times S^1$ by identifying three distinct points to one point. Find an explicit cell structure on $X$."
It is a well-known result that a torus with 2 points identified would be homotopic to a torus wedge sum with a circle. So I followed that proof and intuitively visualize a torus with 3 points identified as 1 would be homotopic to A TORUS WEDGE SUM WITH TWO CIRCLES AT ONE POINT. And hence I think the cell structure would be $(e_0 \cup e_1\cup e_1 \cup e_2) \cup e_1 \cup e_1$ where $(e_0 \cup e_1\cup e_1 \cup e_2)$ is the cell structure of the torus and the two 1-cells represents two circles that maps their boundaries to the $0$-cell of the torus.
Am I right? And as you can see, I mainly use intuition here. So if you can give a vigorous proof, it would be great.