- #1

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if 0 and 1 are accumulation points in [0,1] interval what is the open subset they are in ?

i need explanation about this.....pls...

- Thread starter mbarby
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- #1

- 12

- 0

if 0 and 1 are accumulation points in [0,1] interval what is the open subset they are in ?

i need explanation about this.....pls...

- #2

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An accumulation point of a set S is a point x such that every neighborhood of x contains a point of S (that is not x). Right?

Well, take (0,1), then every neighborhood around 1 contains a point of (0,1), thus 1 is an accumulation point of (0,1). And so is 0.

Likewise, 0 are 1 are accumulation points of [0,1].

- #3

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lets assume 1 is accumulation point in (0,1)

then shouldnt it be contained in an interval like (1-e, 1+e) (e=epsilon)

but we dont have 1+e since it exceeds interval border..

where am i wrong now ?

- #4

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You are considering (0,1) as a subset of the space [itex]\mathbb{R}[/itex], or are you considering (0,1) as a subset of itself?

lets assume 1 is accumulation point in (0,1)

then shouldnt it be contained in an interval like (1-e, 1+e) (e=epsilon)

but we dont have 1+e since it exceeds interval border..

where am i wrong now ?

Anyway, 1 is an accumulation point because every interval (1-e,1+e) around 1 contains a point of (0,1). I don't see what 1+e exceeding the interval boundary has to do with this?

- #5

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(0,1) is the space's itself. so we cant take an interval of (1-e, 1+e) without exceeding 1 by +e.

but what i get from your reaction is that we can take the (1-e, 1+e) interval even if it exceeds the boundary, is that right ?

if so why is that possible ? or isnt there a rule against it ? etc.

thx for the quick explanations by the way.

- #6

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Ah, I think I get it. But if your space is (0,1), then 1 doesn't belong to this space, thus cannot be an accumulation point. In [0,1], 1 is an accumulation point of (0,1). Indeed, the set (1-e,1] is open in [0,1] and contains points from (0,1).

(0,1) is the space's itself. so we cant take an interval of (1-e, 1+e) without exceeding 1 by +e.

but what i get from your reaction is that we can take the (1-e, 1+e) interval even if it exceeds the boundary, is that right ?

if so why is that possible ? or isnt there a rule against it ? etc.

thx for the quick explanations by the way.

- #7

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is it true , then , if i say any interval having the border of the interval is open, or sth similar to that ?

- #8

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That's the problem right there. 1 is an accumulation point OF (0,1). But it's wrong to say that 1 is IN (0,1). I believe your use of "in" rather than "of" is causing you to mis-think about this.what really confuses me here is that:

lets assume 1 is accumulation point in (0,1)

- #9

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Well, yes, something similar. Things like ]a,1] and [0,a[ are open in [0,1]. But things like [0.5,1] isn't...

is it true , then , if i say any interval having the border of the interval is open, or sth similar to that ?

- #10

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that eased my mind. i was tearing myself apart to understand where i was making the mistakes :/ ...

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