# Prob. 4, Sec. 29, in Munkres' TOPOLOGY, 2nd ed: $[0, 1]^\omega$ with uniform topology is not locally compact

Here is Prob. 4, Sec. 29, in the book Topology by James R. Munkres, 2nd edition:

Show that $$[0, 1]^\omega$$ is not locally compact in the uniform topology.

Here is a Math Stack Exchange (MSE) post that is of course relevant. However, here I would like to present my own attempt:

First of all, here is an MSE post of mine on $$[0, 1]^\omega$$ with the uniform topology.

Suppose if possible that $$[0, 1]^\omega$$ with the uniform metric topology is locally compact.

Let $$\mathbf{a} \colon= \left( \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \ldots \right). \tag{Definition 0}$$

As $$[0, 1]^\omega$$ is locally compact at point $$\mathbf{a}$$, so there exists a compact subspace $$C$$ of $$[0, 1]^\omega$$ and an open set $$U$$ in $$[0, 1]^\omega$$ such that $$\mathbf{a} \in U \subset C. \tag{0}$$

Now as $$U$$ is an open set in the uniform metric space $$[0, 1]^\omega$$ and as $$\mathbf{a} \in U$$, so there exists a real number $$\delta > 0$$ such that $$B ( \mathbf{a}, \delta ) \subset U, \tag{1}$$ where $$B ( \mathbf{a}, \delta ) \colon= \{ \, \mathbf{x} \in [0, 1]^\omega \, \colon \, \bar{\rho}( \mathbf{x}, \mathbf{a} ) < \delta \, \}. \tag{ Definition 1 }$$ Since reducing $$\delta$$ will make the set $$B ( \mathbf{a}, \delta )$$ smaller, we can assume without any loss of generality that our $$\delta$$ satisfies $$0 < \delta < \frac{1}{2}. \tag{1*}$$

From (0) and (1) above we also obtain $$B ( \mathbf{a}, \delta ) \subset C. \tag{2}$$

Since $$C$$ is a compact subspace of the Hausdorff space $$[0, 1]^\omega$$ with the uniform metric topology, therefore $$C$$ is also closed in $$[0, 1]^\omega$$, by Theorem 26.3 in Munkres.

Now as $$C$$ is a closed set in $$[0, 1]^\omega$$ and as $$B ( \mathbf{a}, \delta ) \subset C$$ by (2) above, so we also have $$\overline{B ( \mathbf{a}, \delta ) } \subset C,$$ that is, $$\bar{B} ( \mathbf{a}, \delta ) \subset C, \tag{3}$$ where $$\bar{B} ( \mathbf{a}, \delta ) \colon= \{ \, \mathbf{x} \in [0, 1]^\omega \, \colon \, \bar{\rho}( \mathbf{x}, \mathbf{a} ) \leq \delta \, \}. \tag{ Definition 2}$$

Moreover, as $$\bar{B} ( \mathbf{a}, \delta )$$ is a closed set in the compact space $$C$$, so $$\bar{B} ( \mathbf{a}, \delta )$$ is also compact, by Theorem 26.2 in Munkres.

Finally, as $$\bar{B} ( \mathbf{a}, \delta )$$ is a compact (metrizable) space, so it is also limit point compact, by Theorem 28.1 in Munkres.

Now let us take $$\alpha \colon= \frac{1}{2} - \frac{\delta}{2}, \qquad \mbox{ and } \qquad \beta \colon= \frac{1}{2} + \frac{\delta}{2}. \tag{Definition 3*}$$ And, then let us define the set $$A$$ as $$A \colon= \{ \, \alpha, \beta \, \}^\omega. \tag{Definition 3 }$$ Then $$A$$ is an infinite subset of $$\bar{B} ( \mathbf{a}, \delta )$$, but $$A$$ has no limit points in $$\bar{B} ( \mathbf{a}, \delta )$$, as has been shown in my post here. This contradicts the fact that $$\bar{B} ( \mathbf{a}, \delta )$$ is limit point compact.

Thus our supposition at the start of this proof is wrong. Hence $$[0, 1]^\omega$$ in the uniform topology is not locally compact.

Is this proof correct? Is it easy enough to understand? Or, are there issues of accuracy or clarity?

I think the proof is fine, and easy enough. It's a generalisation of the idea to show the unit ball in $$\ell^\infty$$ not being compact.