In Munkres Theorem 20.4 it is shown that the (relative) uniform topology induced by: $$ d(x,y)\triangleq \sup_{n \in \mathbb{N}} d(x_n,y_n) $$ is strictly finer than the product topology on $\prod_{n \in \mathbb{N}} \mathbb{R}$ and its relative topology on $\ell^{\infty}(\mathbb{R})$.

Now, for each positive integer $n$, let $K_k\triangleq \left\{x \in \prod_{n \in \mathbb{N}} \mathbb{R}:\,
|x_n|\leq k
\right\}$. Clearly $K_k \subseteq \ell^{\infty}(\mathbb{R})\subseteq \prod_{n \in \mathbb{N}} \mathbb{R}$ and $K_k \subseteq K_{k+1}$ defines an inductive system in the category **Top**.

How does the inductive limit topology on $\bigcup_{k \in \mathbb{N}} K_k$ compare to (restriction of) the *uniform topology* thereon? My intuition tells me that they coincide, is this correct?

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