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(Mathematics, other places where you need to talk about the real numbers:)
An interval is a subset IR of the real numbers with the property that if x,yI and z is between x and y, then z∈I too.

It's just a "contiguous" segment of the line (which may or may not be bounded on either end). Or, if you like, it's a convex subset of R -- 1-dimensional convexity is boring.

Examples:

Here are an open interval, a closed interval, a half-open interval (which is also half-closed) and 2 unbounded intervals:

(-2,3) = {x: -2<x<3}
[4,7] = {x: 4≤x≤7}
[11,17) = {x: 11≤x<17}
(-,9] = {x: x≤9}
(-∞,+∞) = R
Note: The symbol "∞" in an unbounded interval means precisely nothing: the only meaning here is carried by the entire symbol "(-∞,9]".