Topological spaces and continuity

A topology on a set X is a collection of subsets, declared open, containing both the empty set and X. Arbitrary unions of open sets must be open, as must finite intersections. Arbitrary intersections are not required: the intersection of the intervals $(-1/n,1/n)$ over positive integers is the singleton {0}, which is not open in the usual real line. These axioms abstract the neighborhood behavior of ordinary open intervals without assuming coordinates or a metric.

The same underlying set can carry different topologies. The discrete topology declares every subset open; the indiscrete topology declares only the empty set and the whole space open. A metric generates open sets by unions of open balls, but not every topology comes from a metric. Even when two metrics generate the same topology, their numerical distances can disagree. Topological equivalence preserves continuity, not the length of a route, the size of an angle, or a particular error tolerance.

For a map from X to Y, continuity means that every open subset of Y has an open preimage in X. A preimage exists for any function; it does not require an inverse function. It gathers all inputs whose outputs satisfy the chosen open condition. In metric spaces this criterion is equivalent to the familiar epsilon-delta definition. On the real line, the preimage of an open output interval can have several components and still be open.

Images of open sets need not be open under a continuous map. The constant map from the real line to itself is continuous, but sends every nonempty open interval to a singleton. This is why replacing preimage by image changes the definition. Continuity also depends on both topologies, not just the formula. A map out of a discrete domain is automatically continuous, whereas a map into a discrete codomain must have open fibers over its individual values.

A homeomorphism is a bijection whose forward map and inverse map are both continuous. Bijection alone matches points; homeomorphism also matches the neighborhood structure. Stretching an interval by a positive constant is a homeomorphism onto the stretched interval, although it changes distances. For a useful sufficient condition, a continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Removing these hypotheses requires checking the inverse separately.

Manifolds will add local Euclidean coordinates to topology, and differentiable manifolds will demand smooth transition maps. Those are additional layers rather than consequences of the open-set axioms. A topological argument should therefore state its ambient spaces and avoid silently importing differentiability or metric measurements. When a computer displays only finitely many sample points, it can suggest a continuous shape but cannot verify the behavior of every neighborhood required by the definition.