On my 98 deville a hose running from the coolant top is not connected to anything is it suppose to be like that

There is a hose from the radiator, by the cap, going to the
bottom of the overflow or expansion tank, as you were
referring to

Have you checked to see if your fans are coming on? have you checked for any hose leaks? what about the cap? (sometimes the cap looses the pressure that it is suppose to have) is the passenger floor board getting wet? (heater core leak). Have you flused the system?

You did not mention if you replaced the thermostat. It may be stuck closed.

There is an over-flow hose between the coolant recovery tank, and the top of the radiator just below the radiator cap, and there is also an over-flow hose from the top of the coolant recovery tank and it just hangs over the side of the recovery tank and runs the over-flow off out under the front of the passenger side fender well area.

The most common overheating problem with the Northstar engine is a blocked purge line(small hose).The purge line is designed to purge air out of the cooling system.
There is a small diameter hose coming from near the top of the coolant tank (not the cap) ,this is the purge line.
With the engine idling,pull the purge line from the tank,if there is no coolant flow from this small hose,it is blocked and this will cause overheating.
If the hose is clear, check for blockage at the nipple on the tank.Or trace the hose back until you find the blockage.
The hose goes in to the engine to a crossover and comes out the other side close to the thermostat housing,the blockage may be there.
Also note that the cooling fans don't come on until the coolant temperature hits around 225 degrees.
During normal highway driving the fans will very rarely come on(if at all) due to constant air flow through the radiator.They will only usually come on in heavy traffic conditions on a hot day.
But......if you have the Air Con switched on, the fans will run all of the time.

There is no radiator cap on the Northstar engine.
The only cap is on the surge tank (or expansion tank) and is pressurized.

So the answer is Yes, a faulty surge tank cap will result in the loss of coolant and therefore lead to overheating.

The most common overheating problem with the Northstar engine is a blocked purge line(small hose).The purge line is designed to purge air out of the cooling system.
There is a small diameter hose coming from near the top of the coolant tank (not the cap) ,this is the purge line.
With the engine idling,pull the purge line from the tank,if there is no coolant flow from this small hose,it is blocked and this will cause overheating.
If the hose is clear, check for blockage at the nipple on the tank.Or trace the hose back until you find the blockage.
The hose goes in to the engine to a crossover and comes out the other side close to the thermostat housing,the blockage may be there.

The normal operating temperature for the Northstar engine is around 210 degrees.
The fans will not come on until the temp hits 225 degrees.

The thermostat in the northstar engine is located at the bottom radiator hose (unlike most conventional cooling systems).

The most common overheating problem with the Northstar engine is a blocked purge line(small hose).The purge line is designed to purge air out of the cooling system.
There is a small diameter hose coming from near the top of the coolant tank (not the cap) ,this is the purge line.
With the engine idling,pull the purge line from the tank,if there is no coolant flow from this small hose,it is blocked and this will cause overheating.If the hose is clear, check for blockage at the nipple on the tank.Or trace the hose back until you find the blockage.The hose goes in to the engine to a crossover and comes out the other side to near the thermostat housing,the blockage may be there.

If the other end of your purge line is not connected to anything then that is the problem......it must be reconnected otherwise the cooling system is not pressurized.

The most common overheating problem with the Northstar engine is a blocked purge line. It's job is to purge air out of the cooling system.
There is a small diameter hose coming from near the top of the coolant tank,this is the purge line.
With the engine idling,pull the purge line from the tank,if there is no coolant flow from this small hose,it is blocked and this will cause overheating.If the hose is clear, check for blockage at the nipple on the tank.

do the heater core coolant supply hoses both get hot at the firewall in the engine compartment? if yes then you have a defective temperature control blend door actuater motor.

The most common overheating problem with the Northstar engine is a blocked purge line. It's job is to purge air out of the cooling system.
There is a small diameter hose coming from near the top of the coolant tank,this is the purge line.
With the engine idling,pull the purge line from the tank,if there is no coolant flow from this small hose,it is blocked and this will cause overheating.If the hose is clear, check for blockage at the nipple on the tank.