Is unit circle a simplicial complex?

Question

If so , please help me to find the simplices.

I think it is homotopic to a point .so there is only one vertex.

Answer 1

If $e_1,e_2$ denote the two standard basis vectors of $\Bbb R^2$, then the sets $$A\subset\{e_1,-e_1,e_2,-e_2\},\ |A\cap\{e_i,-e_i\}|\le 1 \text{ for all }1\le i\le 2$$ are the simplices of a simplicial complex $X$ homeomorphic to $S^1$. In fact, the realization $|X|$ is the set $$\{(x_1,x_2)\in\Bbb R^2\mid |x_1|+|x_2|=1\}$$ and $x\mapsto\dfrac x{\ ||x||_2}$ is a homeomorphism to the unit circle.