seriously disabled Posted April 19, 2009 Share Posted April 19, 2009 An open interval is an interval that does not include its end points. http://mathworld.wolfram.com/OpenInterval.html What I don't understand is why is it called an open interval? Link to comment Share on other sites More sharing options...

PhDP Posted April 19, 2009 Share Posted April 19, 2009 It's all about sets; As an example, consider the open interval (0, 1) consisting of all real numbers x with 0 < x < 1. Here, the topology is the usual topology on the real line. We can look at this in two ways. Since any point in the interval is different from 0 and 1, the distance from that point to the edge is always non-zero. Or equivalently, for any point in the interval we can move by a small enough amount in any direction without touching the edge and still be inside the set. Therefore, the interval (0, 1) is open. However, the interval (0, 1] consisting of all numbers x with 0 < x ≤ 1 is not open in the topology induced from the real line; if one takes x = 1 and moves an arbitrarily small amount in the positive direction, one will be outside of (0, 1']. Link to comment Share on other sites More sharing options...

Shadow Posted April 19, 2009 Share Posted April 19, 2009 If you're asking about the reason behind the name, then I'm not sure, although I guess it's because it "goes on for ever" (ie.: has no minimum or maximum) therefore it's not "closed", but "open". EDIT: Woops, didn't notice the post above ) Link to comment Share on other sites More sharing options...

seriously disabled Posted April 19, 2009 Author Share Posted April 19, 2009 If you're asking about the reason behind the name, then I'm not sure, although I guess it's because it "goes on for ever" (ie.: has no minimum or maximum) therefore it's not "closed", but "open". EDIT: Woops, didn't notice the post above ) See http://www.mathwords.com/o/open_interval.htm But in this example it does. In this example the minimum is -2 and the maximum is 3. So why is it really called an open interval? Link to comment Share on other sites More sharing options...

Shadow Posted April 19, 2009 Share Posted April 19, 2009 Here is what I mean: As an example, consider the open interval (0, 1) consisting of all real numbers x with 0 < x < 1. Here, the topology is the usual topology on the real line. We can look at this in two ways. Since any point in the interval is different from 0 and 1, the distance from that point to the edge is always non-zero. Or equivalently, for any point in the interval we can move by a small enough amount in any direction without touching the edge and still be inside the set. Therefore, the interval (0, 1) is open. However, the interval (0, 1] consisting of all numbers x with 0 < x ≤ 1 is not open in the topology induced from the real line; if one takes x = 1 and moves an arbitrarily small amount in the positive direction, one will be outside of (0, 1']. Link to comment Share on other sites More sharing options...

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