Questioning about the meaning of "$1$-dimensional circle"

Question

When we talk about the $1$-dimensional circle, is it a one-dimensional object, although one can embed it into a two-dimensional object? More precisely, is it a one-dimensional manifold?

Answer 1

Yes, it is a one-dimensional manifold (i.e. it locally looks like $\mathbb R$).

The fact that you can embed it into a higher dimensional space makes no difference, since you can also embed $\mathbb R^3$ into $\mathbb R^4$, but it's still a three dimensional object.