Editing Local property (section)

==Properties of a pair of spaces==

Given some notion of equivalence (e.g., [[homeomorphism]], [[diffeomorphism]], [[isometry]]) between [[topological space]]s, two spaces are said to be locally equivalent if every point of the first space has a neighborhood which is equivalent to a neighborhood of the second space.  

For instance, the [[circle]] and the line are very different objects.  One cannot stretch the circle to look like the line, nor compress the line to fit on the circle without gaps or overlaps.  However, a small piece of the circle can be stretched and flattened out to look like a small piece of the line.  For this reason, one may say that the circle and the line are locally equivalent.

Similarly, the [[sphere]] and the plane are locally equivalent. A small enough observer standing on the [[Surface (topology)|surface]] of a sphere (e.g., a person and the Earth) would find it indistinguishable from a plane.