Does the water level rise? if I have A closed vessel that is half-filled with water. There is a hole near the top of the vessel and air is pumped out from this hole.
will the level of water rise

apparently, the level of water doesn't always rise
I thought since there is no atmospheric pressure:
$$  P_{atm} + DgH_1=DgH_2$$
thus causing the level to rise
 A: The water level will not rise, it will actually gradually fall. This is because since there is no air pressure, the water molecules can easily escape the liquid and turn into water vapor which gradually fills the vacuum. If the container is sealed after creating the vacuum, it will fall until a certain level and then stop, because it will create an equilibrium where the number of water molecules evaporating from the liquid water will be equal to the number of water vapor molecules reentering the liquid water. If the container is not sealed then it will keep falling until there is no liquid water left.
A: Are you thinking of a U shaped tube, where if you decrease the pressure on one side, pressure on the other forces the fluid to rise?

Image from Physics Libretexts
You can see how pressing on one side would make the fluid move. In the case you have shown, pressing on the top of the fluid just squeezes it.
A: What you will see happen is the water will start to boil.
A: There are actually some counter-acting phenomena at play here.

*

*No compressibility: in a first approximation, the water can be considered as incompressible. Then we come to the conclusion that the water starting to vaporize for reduced vessel pressure.


*More in detail, the water density changes with temperature, and the evaporation will cool the water down, which will increase its density, and hence lower the water level further (at least, if it was initially above approx 4degC). if it was below 4degC, then cooling it will initially decrease the density.


*Also, water isn't completely incompressible, and lowering the vessel pressure will lead to an (albeit small) expansion. The question whether this expansion will
outweigh the vaporization depends a lot on the shape of the vessel. Say, the effects could be strong enough, if you have a long thin tube, that is only open on one end, like this (yes, we're maintaining picture quality here.. :) ) :

