Open sets on a circle

Question

Imagine a one-dimensional being living on a circle. For him space is given by an oriented (for example clockwise) coordinate $x$ on the circle, and the being sees the points of the space as number 0<=x<(circumference of the circle). Since he wants to use the topological manifold structure for describing the space where he lives he has to define what is an open set on the circle. But how can he define what is an open set for the circle? because with the standard definition he can not find an open set around $x=0$. It would be better an answer with little math since I'm more interested in the concept

Answer 1

An open set is technically a closed set, taken out the boundary. In your case, for simplicity let us assume $1$ be the circumference of the circle. Then $(0,1)$, the set of all points from $0$ to $1$ excluded the two points at the ending is one such open set. Hope that helps :)

Answer 2

Say the origin $0$ is the center of the circle of radiues $r$, say1. Since $0$ does not belong to the circle, it is perfectly normal that no open set of the circle contains it.

A basis of of open sets (for the most natural topology, namely the topology induced by the topology of $\mathbb{R}^2$) is given by the $U_{\alpha, \beta}$s, $\alpha, \beta \in [0; 2\pi]$, where :

$$U_{\alpha, \beta} := \{ P \in \mathbb{R^2} \ \big| \ P \textrm{ has polar coordinates } (r,\theta) \textrm{ with } \alpha < \theta < \beta\}$$

In other words, the basic open sets of the circle are "open arcs".