In mathematics, something is said to occur locally in the category of topological spaces if it occurs on "small enough" open sets.

Binary relation

Given some notion of equivalence (e.g., homeomorphism, diffeomorphism, isometry) between topological spaces, two spaces are 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 are the plane are locally equivalent. A small enough observer standing on the surface of a sphere (e.g., a person and the Earth) would find it indistinguishable from a plane.

Unary relation

If P Is a property of topological spaces, then a space is sometimes said to be "locally P" if every point of the space has a neighborhood system of sets with property P. Such is the case with

• Locally compact topological spaces
• Locally connected and Locally path-connected topological spaces

An exception is a locally closed subset of a topological space, which is simply the intersection of an open set and a closed set.