I am also confused about this issue of open vs. closed intervals, so I figured I'd try to answer this fine question to see if I could straighten out my own thinking.
One way to look at the property of being an increasing function is that if you go from a up to b, the function value also increases. That is, a function is increasing on an interval if for any a < b in the interval, f(a) < f(b) (or maybe = if not strictly increasing...) Now, that's true on a closed interval including the max and min of the function; that is, for sin, on [-pi/2, pi/2] it has this property.
On the other hand, when you think of increasing not as a property of two DIFFERENT points in the interval, but rather as a property happening AT a single point, then certainly sin is not increasing at pi/2, its derivative is 0 there and it is a maximum point. You can't increase past a maximum! So in this interpretation, where we FIRST classify each individual point according to what's going on locally there and THEN group them into intervals based on that, the set of increasing points would be a union of open intervals like (-pi/2, pi/2).