General Topology/Continuity

Continuity is a local property, in that it may be characterized by a property that a function might have at every point.

When $Y$ is a uniform space, the definition of equicontinuity simplifies, and furthermore in this situation equicontinuous subsets are related to compact subsets of $𝒞(X,Y)$. This we will see in the chapter on uniform structures.

In other words, a function $f:X\to Y$ is a local homeomorphism if and only if for all $x_0\in X$, there exists an open neighbourhood $U$ of $x_0$ and an open neighbourhood $V$ of $f(x_0)$ so that $f{|}_U$ is a homeomorphism from $U$ to $V$.