Compact-open topology

is sufficient. It permits functions such as translated compactly supported bumps to converge to zero on every compact set while retaining the same global supremum norm. The compact-open topology records local approximation rather than a bound over an unbounded domain all at once.

Table of contents
- Weighted metric for local uniform convergence Compact-open topology

Weighted metric for local uniform convergence ( $σ (f, g) = \sum_{n} w_{n} ϕ (∥ f - g ∥_{K_{n}})$ )

 0  0

Let

X

have nested compact sets

K_{n}

covering it, with

K_{n} \subseteq int K_{n + 1}

. Then every point has a neighborhood in some

K_{n}

, and every compact subset is contained in one

K_{n}

. For continuous scalar functions, take positive weights with

\sum_{n} w_{n} < \infty

and define

σ (f, g) = \sum_{n = 1}^{\infty} w_{n} \frac{∥ f - g ∥ _{K_{n}}}{1 + ∥ f - g ∥ _{K_{n}}} .

(1)

The bounded metric transform makes the sum finite and gives its triangle inequality; the covering property gives definiteness. On

R

, one may use

K_{n} = [- n, n]

and

w_{n} = 2^{- n}

σ (f_{j}, f) \to 0

, each fixed compact supremum norm tends to zero, since its summand is bounded by the whole metric. Conversely, control finitely many compact supremum norms and then the summable weight tail. This proves that

σ

realizes locally uniform convergence under the stated exhaustion conditions.

Merely writing a space as a union of compact sets is insufficient. Enumerate the points of the compact space

{0} \cup {1/ n : n \geq 1}

and take finite initial subsets as

K_{n}

. The resulting metric detects pointwise control on those finite sets, not uniform control on the entire compact space. The neighborhood and compact-containment conditions explain the difference.