From reading some introductory texts on differential geometry, the author would usually invoke the phrase "locally looks like" when it comes to defining a manifold.
For example, the real line is a manifold because it locally looks like a real line.
Can someone provide a slightly more mathematical definition of "looks like"? Does the real line looks like the entire real line or some segment of it? What about a circle?
A space $M$ is locally euclidean of dimension of $n$ if for any point $p\in M$, there is an open neighborhood $U$ of $p$ and a map $\varphi: U\to \varphi(U)\subset \mathbb{R}^n$ which is a homeomorphism. These are called charts of $M$.
We usually also want some sort of smoothness condition, which is usually stated as having the transition maps $\psi\circ\varphi^{-1}:\varphi(U\cap V)\to \psi(U\cap V)$ are smooth as functions on $\mathbb{R}^n$, where $\psi:V\to\psi(V)$ is defined as above.
This is usually what is meant when an author writes "looks locally like" Euclidean space.