Accumulation Points in [0,1]: Understanding the Boundaries

  • Context: Graduate 
  • Thread starter Thread starter mbarby
  • Start date Start date
  • Tags Tags
    Point
Click For Summary

Discussion Overview

The discussion revolves around the concept of accumulation points, specifically whether the points 0 and 1 are accumulation points in the intervals (0,1) and [0,1]. Participants explore the definitions and implications of accumulation points within these contexts, raising questions about the nature of open and closed intervals in topology.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that 0 and 1 are accumulation points of the interval (0,1) because every neighborhood around these points contains points from (0,1).
  • Others argue that since 1 is not contained in (0,1), it cannot be an accumulation point of that interval.
  • A participant expresses confusion about the definition of accumulation points and the implications of using intervals that exceed the boundaries of the set.
  • Another participant clarifies that while 1 is not in (0,1), it can still be considered an accumulation point of (0,1) when viewed in the context of the larger set [0,1].
  • There is a discussion about the nature of open intervals and whether intervals that include boundary points can be considered open.
  • Some participants reflect on their understanding of topology and express gratitude for the explanations provided.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether 0 and 1 are accumulation points in (0,1), with some asserting they are while others maintain that they are not due to the definitions involved. The discussion remains unresolved regarding the implications of boundary points in open intervals.

Contextual Notes

There are limitations in the discussion regarding the definitions of open and closed sets, as well as the assumptions made about the nature of neighborhoods in relation to accumulation points.

Who May Find This Useful

This discussion may be useful for individuals interested in topology, mathematical definitions, and the nuances of set theory, particularly those who are new to the concepts of accumulation points and interval properties.

mbarby
Messages
12
Reaction score
0
are 0,1 accumulation points in (0,1) ? how about in [0,1] ?
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...
 
Physics news on Phys.org
Hi mbarby! :smile:

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].
 
what really confuses me here is that:
lets assume 1 is accumulation point in (0,1)
then shouldn't it be contained in an interval like (1-e, 1+e) (e=epsilon)
but we don't have 1+e since it exceeds interval border..
where am i wrong now ?
 
mbarby said:
what really confuses me here is that:
lets assume 1 is accumulation point in (0,1)
then shouldn't it be contained in an interval like (1-e, 1+e) (e=epsilon)
but we don't have 1+e since it exceeds interval border..
where am i wrong now ?

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?

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?
 
sorry for the questions , i am not a math guy. but topology is one of the topics i want to learn. so somethings are as clear to me as it is to you guys.
(0,1) is the space's itself. so we can't 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 isn't there a rule against it ? etc.
thx for the quick explanations by the way.
 
mbarby said:
sorry for the questions , i am not a math guy. but topology is one of the topics i want to learn. so somethings are as clear to me as it is to you guys.
(0,1) is the space's itself. so we can't 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 isn't there a rule against it ? etc.
thx for the quick explanations by the way.

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).
 
thx a lot this explains a great to me.
is it true , then , if i say any interval having the border of the interval is open, or sth similar to that ?
 
mbarby said:
what really confuses me here is that:
lets assume 1 is accumulation point in (0,1)

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.
 
mbarby said:
thx a lot this explains a great to me.
is it true , then , if i say any interval having the border of the interval is open, or sth similar to that ?

Well, yes, something similar. Things like ]a,1] and [0,a[ are open in [0,1]. But things like [0.5,1] isn't...
 
  • #10
thx all guys,
that eased my mind. i was tearing myself apart to understand where i was making the mistakes :/ ...
 

Similar threads

Undergrad (0,1) is uncountable using binary expansions?
  • · Replies 15 ·
Replies
15
Views
4K
Undergrad Is [0,1] under the subspace topology Hausdorff, Compact, or Connected?
  • · Replies 15 ·
Replies
15
Views
4K
Undergrad Why is [0,1] compact in the case of a collection of open neighborhoods?
  • · Replies 9 ·
Replies
9
Views
6K
Graduate Question about existence of path-lifting property
  • · Replies 8 ·
Replies
8
Views
3K
High School Is the Explanation of Compactness in [0,1] and (0,1) Accurate?
  • · Replies 24 ·
Replies
24
Views
3K
MHB What Are the Accumulation Points of the Sequence $1/n$?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Question about lifting of a function
  • · Replies 2 ·
Replies
2
Views
3K
MHB Using the Definition: Showing $u \in C^0 [0,1]$
  • · Replies 2 ·
Replies
2
Views
2K
Understanding Equlibrium Points in differential equations
  • · Replies 9 ·
Replies
9
Views
2K
Graduate Is this function in the L^2(0,1) space for certain values of C_1 and C_2?
  • · Replies 6 ·
Replies
6
Views
2K