Real Analysis. Definition of open sets. I'm trying to understand it in terms of logical statement. So the definition is: a set E in a metric space X is open if every point of E is an interior point. So what I came up with is ∀x∈E ∃r0 ∀q ( d(x,q)

Question

Real Analysis. Definition of open sets. I'm trying to understand it in terms of logical statement. 

So the definition is: a set E in a metric space X is open if every point of E is an interior point. 

So what I came up with is ∀x∈E ∃r>0 ∀q ( d(x,q) < r → q∈E). Does this sound right?

ganeshie8 · Answer

basically you're defining an openball around each point in the set

ganeshie8 · Answer

looks good to me

anonymous · Answer

yeah, not just any open ball. An open ball around x such that every points in the ball must be in E.

tkhunny · Answer

...and you are NOT including the boundary.

anonymous · Answer

the boundary of which set? @tkhunny

ganeshie8 · Answer

Oh right, lets see if your definition meets below :

|dw:1416634811160:dw|

ganeshie8 · Answer

define the set as all the points interior to that region except that one middle point

is it open ?

anonymous · Answer

are you saying the middle point is not in the set? @ganeshie8

ganeshie8 · Answer

yes

anonymous · Answer

I think it's open|dw:1416635278456:dw|

although that middle point is causing some concert because. d(x,x) = 0 < r but x is not in the set E. So there is flaw in my definition?