📂Topology

Function Spaces in Topology

Definition¹

For topological spaces $X$ and $Y$, the product space $Y^{X}$ defined as follows is called a function space. $$ Y^{X} : = \prod_{x \in X} Y = \left\{ f \ | \ f : X \to Y \text{ is a function} \right\} $$

The following are topologies on the function space:

1. For $x \in X$ and an open set $U$ in $Y$, let $$ S (x , U) = \left\{ f \in Y^{X} \ | \ f(x) \in U \right\} $$ The topology on $Y^{X}$ generated by the subbasis $\left\{ S(x,U) \ | \ x \in X , U \subset Y \right\}$ is called the point-Open topology.
2. For a compact set $K \subset X$ and an open set $U$ in $Y$, let $$ S (K , U) = \left\{ f \in Y^{X} \ | \ f(K) \subset U \right\} $$ The topology on $Y^{X}$ generated by the subbasis $\left\{ S(K,U) \ | \ K \subset X , U \subset Y \right\}$ is called the compact-Open topology.

Suppose $(Y,d)$ is a metric space.

3. For a compact set $K \subset X$ and $\varepsilon > 0$, let $$ B_{K} (f, \epsilon) = \left\{ g \in Y^{X} \ \left| \ \sup_{x \in K} \left\{ d(f(x),g(x)) \right\} < \varepsilon \right. \right\} $$ The topology on $Y^{X}$ generated by the basis $\left\{ B_{K} (f, \varepsilon ) \ | \ K \subset X , \varepsilon > 0 \right\}$ is called the topology of Compact Convergence.
4. The uniform metric $$ \overline{ \rho } (f,g) : = \sup_{x \in X} \left\{ \min \left\{ d(f(x) , g(x) ) , 1 \right\} \right\} $$ generates the metric space $(Y^{X} , \overline{ \rho } )$, and its topology is called the uniform topology.

Theorem

• [1]: The compact-Open topology is larger than the point-Open topology.
• [2]: The topology of Compact Convergence is larger than the point-Open topology.
• [3]: The uniform topology is larger than the compact-Open topology.
• [4]: The uniform topology is larger than the topology of Compact Convergence.
• [5]: If $X$ is a discrete space, the topology of Compact Convergence on $Y^{X}$ is the same as the point-Open topology.
• [6]: If $X$ is a compact space, the topology of Compact Convergence on $Y^{X}$ is the same as the uniform topology.

Let $\left\{ f_{n} : X \to Y \right\}$ be a sequence in $Y^{X}$, and let the function whose domain is restricted to $K \subset X$ be denoted by $f_{n} |_{K} : K \to Y$.

• [7]: $\left\{ f_{n} \right\}$ converges to $f$ in the point-Open topology of $Y^{X}$. $\iff$ For every $x \in X$, $ f_{n} (x) $ converges to $f(x)$.
• [8]: $\left\{ f_{n} \right\}$ converges to $f$ in the topology of Compact Convergence of $Y^{X}$. $\iff$ For every compact $K \subset X$, $f_{n} |_{K}$ converges uniformly to $f |_{K}$.

Let the set of continuous functions whose domain is a topological space $X$ and whose codomain is a metric space $Y$ be $$ C(X,Y) := \left\{ f \in Y^{X} \ | \ f \text{ is continuous} \right\} $$ and let $C(X,Y)$ be a subspace of $Y^{X}$.

• [9]: The compact-Open topology and the topology of Compact Convergence on $C(X,Y)$ are the same.
• [10]: The topology of Compact Convergence on $C(X,Y)$ does not depend on the metric of $Y$.
• [11]: If a sequence $\left\{ f_{n} \right\}$ in $C(X,Y)$ converges to $f \in Y^{X}$, then $f : X \to Y$ is a continuous function.

Explanation

In particular, $C(X, \mathbb{R})$ is denoted by $C(X)$, and especially when $X$ is an interval, that is, when $X=(a,b)$ or $X=[a,b]$, it is also denoted by $C(a,b)$ or $C[a,b]$, respectively.

[1]~[4]

Roughly summarizing, one can say that the point-Open topology tends to be small and the uniform topology tends to be large.

[7], [8]

These can be usefully applied to show that a function is uniformly continuous.

[10], [11]

When general topology is viewed as a generalization of analysis, these are very important as facts.