Reinforcement leaning, what are bounded and ubounded action space mean? Current policy gradient methods usually focus on MDPs with unbounded action spaces. So, here what are the bounded and unbounded action space mean? 
For bounded action space, is it mean there are limited actions for agents to choose from? And For unbounded action space, is it mean there are infinite actions,for example, actions take from a range $(0, 1]$?
 A: Bounded and unbounded have slightly different meanings from finite/infinite.
A bounded action space means that there are clearly defined limits / bounds on the space of actions, and there are no legal actions outside that space.
An unbounded action space means the opposite, that the action space is not entirely bounded.
Having an unbounded action space does indeed seem to imply that the action space has an infinite size, I can't really imagine an unbounded action space that is finite.
Bounded action spaces can be finite or infinite though. Two examples:


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*Finite, bounded action space: $\mathcal{A} = \{ 0, 1, 2 \}$, a discrete action space with 3 different actions.

*Infinite, bounded action space: $\mathcal{A} = [0, 1)$. This is the example you gave for what you thought was unbounded, but this is not unbounded, it's bounded; there are clear bounds $0$ and $1$. It's still infinite, because there are infinitely many numbers between $0$ and $1$.
Two examples of unbounded action spaces (both infinite):


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*$\mathcal{A} = \mathbb{R}$ (all the real numbers)

*$\mathcal{A} = \{0, 1, 2, \dots \}$ (all nonnegative integers)


Note that the second example here still has one "bound" in $0$, but has no upper bound, so it's still called unbounded (it might also be called "bounded from below").
