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## Homework Statement

For some reason, the uniform topology always causes me problems. So, let's work this through.

Let Rω be given the uniform topology, i.e. the topology induced by the uniform metric, which is defined with d(x, y) = sup{min{|xi - yi|, 1}, i is in ω}.

Given some n, let Bn be the collections of subsets of Rω of the form ∏Ai, where Ai = R for i <=n, and Ai = {0} or Ai = {1}, for i > n. One needs to show that the collection B = U Bn is countably locally finite, but neither countable nor locally finite.

## The Attempt at a Solution

First of all, it is obvious that B is not countable, since the collection of sequences of zeros and ones is not countable. Let's leave the local finiteness of B for later.

To see if B is countably locally finite, consider Bn, for some fixed positive integer n. Let x be an element of Rω. We need to show that there exists a neighborhood of x which intersects Bn in finitely many members.

Since we're dealing with a metric space, we can consider open balls around the element x in the uniform metric d. But now I'm a bit confused and would be grateful for a push in the right direction.

Thanks in advance, as always.