---
tag: D8GHJN61MW
kind: definition
name: Topology of pointwise convergence
msc: [54C35, 54B10]
defines: FVGB8ZP67B
uses: [0CQHFPGJ45, 0GBG8B49AA, 09S8DV34Y0, 02BH3MCBQB, 0GFGK40582, 0WS9P12FQR]
sources:
  - url: https://en.wikipedia.org/wiki/Pointwise_convergence
---

## Definition

Let $X$ be a set, let $Y$ be a [[topological space|0CQHFPGJ45]], and let $Y^X$ be the set of all maps from $X$ to $Y$.
For $x \in X$ and an [[open|02BH3MCBQB]] set $U \subseteq Y$, put
$$W(x, U) = \set{f \in Y^X : f(x) \in U}.$$
The *topology of pointwise convergence* on $Y^X$ is the topology generated by the [[subbasis|09S8DV34Y0]] of all sets $W(x, U)$.
On a subset $F \subseteq Y^X$, such as the set of continuous maps when $X$ is a space, it is the [[subspace topology|0WS9P12FQR]].

## Remarks

Viewing $Y^X$ as the product $\prod_{x \in X} Y$, this topology is exactly the [[product topology|0GBG8B49AA]].
A sequence $(f_n)$ converges to $f$ in it if and only if $f_n(x) \to f(x)$ in $Y$ for every $x \in X$, which is [[pointwise convergence|0GFGK40582]].
The same holds for nets.

Example: on $\R^{[0, 1]}$, let $f_n(x) = x^n$, and let $f(x) = 0$ for $x < 1$ and $f(1) = 1$.
For each fixed $x$, the numbers $x^n$ tend to $f(x)$, so $f_n \to f$ in this topology.

Non-example: the discrete topology on $\R^{[0, 1]}$ is not the topology of pointwise convergence.
In the discrete topology, the sequence $f_n$ above does not converge, because $\set{f}$ is open and contains no $f_n$.

For an uncountable set $X$ and $Y = \R$, the space $\R^X$ with this topology is not first-countable, so it is not metrizable.