29.The space R' is complete but
not
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The compact is the correct answer.
Step-by-step explanation:
- R is neither compact nor sequentially compact.
- That it isn't always sequentially compact follows from the truth that R is unbounded and Heine-Borel.
- To see that it isn't always compact clearly, notice that the open cowl consisting precisely of the units Un = (−n, n) can have no finite subcover.
- R is entire however now no longer compact. In an area with the discrete metric, the simplest Cauchy sequences are the ones that might be consistent from a few points on.
- Hence any discrete metric area is entire. Thus, a few bounded entire metric areas aren't compact.
- In truth, a uniform area is compact if it is entire and pre-compact. So in a finite-dimensional normed area, each nonbounded closed subset works.
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