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?

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'].

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 )

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?

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'].