meaning given -Region: Open collection with none, some, or all of its boundary points.

This seems quite unimportant... And it seems that almost all sets are areas (I can only think that regions, i can"t think of any type of example the isn"t.).It feels choose it I"m given a set that is, neither open or closed, the could constantly be decomposed by acquisition away the boundary, meaning it is an open set with some of its border points.This is why ns think the following set is a region:$$S = z=x+iy in usmam.orgbbC:xgeq 0, y>0.$$

If the is a region, what would be an example for a non-region?


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edited Jul 25 "17 in ~ 14:16
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inquiry Jul 25 "17 in ~ 14:00
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$egingroup$

There is no general definition for this term and one need to refer to the context concerning its an accurate meaning.

As terry Tao points out in among his lecture note on complex analysis:

The concept of a non-empty open associated subset $U$ the the complicated plane come up so typically in complex analysis that numerous texts entrust a one-of-a-kind term come this notion; because that instance, Stein-Shakarchi refers to such sets together regions, and also in other texts they may be called domains.

The meaning you offered "Open set with none, some, or all of its border points" is seldom used. Come give an example which is no a region under this meaning is fairly easy: just take into consideration a set with only a solitary point on the complicated plane.


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edited Jul 25 "17 in ~ 14:39
reply Jul 25 "17 at 14:14
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$egingroup$ Well, yes, over there is no general meaning and its meaning depends top top the context. $endgroup$
–user9464
Jul 25 "17 at 14:20


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$egingroup$ same enough. Example added. Because that me, this is nothing yet a concept for convenience of act analysis. Ns doubt that civilization in practice would yes, really care around classifying which sets space "region" and also which are not. $endgroup$
–user9464
Jul 25 "17 at 14:52
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$egingroup$
Consider the set of facility numbers: $ x+iy : x = y $ (the line $y=x$ in the 2D plane). This set cannot it is in expressed as an open set with few of its boundary points, since it has no inner points. It is therefore not a region according to your definition.

You are watching: What does region mean in math

More generally, by her definition, any kind of (non-empty) collection with no inner points is not a region.


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edited Jul 25 "17 at 15:05
answer Jul 25 "17 in ~ 14:22
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$egingroup$ does the empty set have an interior? $endgroup$
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Jul 25 "17 in ~ 14:46


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