What is an intuitive way to understand covers and finite subcovers Here are the

Question

What is an intuitive way to understand covers and finite subcovers

Here are the definitions of covering and subcovers

Here is how I inutitively see it using a diagram/graph

Is this the right way to intuitively understand covering and subcovering?

Caverng of a Set in a metic S Pace ?,J)-bea metric-Space-and let Let 4 Coveins oF SS colection of Sets F from X Such that SCK AEF AcallesCovs of S A Solcovering of S is a Sulce lectien of sets SCF Such that AES An oven Covering af S is a Colitectkien of oven sets F from X Such that ACF An OPen Sain of 5 Callectien of aPen sets S F sa that S( AES pace CR.d) whore d is the Euclides S (G.)R, One Sech Co exl consider the metric Stace CR.d) where d is the Euclide CaNCider He Subset

Explanation / Answer

Yes, a subcover is just a subset of the cover such that this subset still covers the entire space. For example, {(0,1),(1/2,2),(1/2,3/2)} will be an open cover of (0,2). A subcover would be {(0,1),(1/2,2)}. A subcover would NOT be {(0,1),(1/2,3/2)}. That said, all the elements of an open cover are open. Thus a subcover of an open cover is again an open cover. If the cover is not necessairily open, then the subcover doesnt need to be open to (but it can be).

Yes, [0,1] may have infinitely many open subsets. But the point is, that among these subsets, you will find finitely many which will still cover [0,1]. Take (0,1), an open cover of this is {(1/2,1),(1/4,1),(1/8,1),...}.

But this does not have an open subcover (thus we cannot find finitely many elements in this cover which still cover (0,1) ). The example you should start of with is the difference between

{ 0 } ? { 1 / n | n > 0 }

and { 1 / n | n > 0 } .

We can find an open cover of the second set which does not have an open subcover (try to find one!). But we cannot do this for the first set (even though it has more elements!). Indeed, take an open cover {G_i}_i of the first set. Then there must exist an i such that G_i contains 0. But since 0 is the limit of the sequence, there must be an N, such that

1/n?Gi

for all i>N. So all but finitely many indices remain uncovered. But we can easily cover these points by finitely many sets. Thus we obtain an finite subcover.

so for me, it is the right way to intuitively understand covering and subcovering

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