Does pointwise equicontinuous and uniformly equicontinuous implies compactness?

Question

If every sequence of pointwise equicontinuous functions $M \rightarrow \mathbb{R}$ is uniformly equicontinuous, does this imply that $M$ is compact?

score 5 · Accepted Answer · edited Feb 16 '16 at 07:59

5

Consider the subspace $M=\mathbb{Z}$ of the reals, an infinite discrete space, certainly not compact. But every sequence of functions is uniformly equicontinuos in that space.

edited Feb 16 '16 at 07:59

Daniel Fischer

212,535

answered May 25 '13 at 16:04

Vobo

3,055

Does pointwise equicontinuous and uniformly equicontinuous implies compactness?

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