Manifolds 7 | Continuity [dark version]

Continuity in topology is defined purely through open sets: a function between topological spaces is continuous exactly when the pre-image of every open set is open. That single rule replaces the familiar ε–δ and sequence-based definitions from real analysis, because topological spaces may lack a metric for measuring “closeness.” Concretely, if there is a map F from a space X with topology TX to a space Y with topology TY, then continuity means that for every open set U in TY, the set F^{-1}(U) must belong to TX. In this framework, “points near each other” is translated into “open sets pull back to open sets,” so the topology itself carries the notion of neighborhood.

The same open-set criterion also clarifies what it means for a map to preserve topological structure in both directions. A homeomorphism is a bijection F: X → Y whose forward map is continuous and whose inverse F^{-1}: Y → X is also continuous. When such a map exists, the spaces are topologically the same: they may look different, but their open-set structure—and therefore their continuity properties—match.

Several examples show how the definition behaves at the extremes. With the indiscrete topology on Y (only ∅ and Y are open), every function F: X → Y is automatically continuous, because the only open sets to check have pre-images that are either empty or all of X. With the discrete topology on X (every subset is open), every function from X to any Y is also continuous, since pre-images of open sets in Y are always open in X. These cases highlight that continuity depends on the chosen topologies, not just on the underlying sets.

A more structural example comes from quotient spaces. Start with a topological space X and an equivalence relation ~ on it. The quotient space Y is formed by collapsing points in the same equivalence class, and its topology is defined so that the canonical projection q: X → Y is continuous. In other words, the quotient topology is engineered precisely to make q continuous.

The transcript then revisits continuity through sequences, connecting topology back to analysis. A function is sequentially continuous if whenever a sequence (x_n) converges to x in X, the sequence of images (F(x_n)) converges to F(x) in Y—so limits can be “pulled through” the function. In general topological spaces, sequential continuity does not always match open-set continuity; however, open-set continuity is stronger, implying sequential continuity. For metric spaces—and more broadly for first countable spaces—the two notions become equivalent, and the same equivalence holds for second countable spaces as well. That equivalence is positioned as a useful tool for later work, before the discussion pivots to compactness in the next installment.