DUAL TRAINING FLT TAXIED OUT WITHOUT TWR CLRNC DUE TO BELIEVING THAT THE TWR WAS STILL CLOSED.

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AFTER STARTING THE ACFT; MY STUDENT TUNED THE COM RADIO TO PICK UP ATIS. THE INFO THAT WE RECEIVED WAS THE ADVISORY THAT RHV TWR USES WHEN IT CLOSES AT NIGHT; IE; 'REID-HILLVIEW TWR SUSPENDED OPS AT XX00Z;' ETC. THINKING THAT THE TWR WAS NOT OPEN AT THE PRESENT TIME; AROUND XX10 LCL; WE TAXIED TO THE RUN-UP AREA FOR RWY 31R WITHOUT MAKING A CALL BUT MONITORING THE CTAF ON 119.80. WE WERE IN THE RUN-UP AREA FOR AROUND 5 MINS WHEN WE WERE READY TO DEPART. MY STUDENT ADVISED THE AREA TFC THAT WE WOULD BE TAKING THE ACTIVE RWY 31R FOR A R DOWNWIND DEP. BEFORE WE TAXIED ONTO THE RWY WE DID A 360 DEG TURN TO SCAN FOR TFC. ONCE WE HAD CLRED THE AREA; WE BEGAN TO TAXI ONTO THE RWY. JUST THEN RHV TWR CAME ON FREQ AND ASKED IF WE WOULD LIKE A WIND CHK BEFORE WE TOOK OFF. WE RESPONDED AND SAID THAT WE WOULD LIKE THE WIND CHK. TWR THEN CAME ON AND SAID 'WIND CALM.' A FEW SECONDS LATER RHV TWR CAME BACK ONTO THE FREQ AND SAID; 'SKYHAWK WOULD YOU LIKE A CLRNC BEFORE YOU TAKE OFF?' MY STUDENT AND I WERE BOTH SHOCKED BY THIS COMMENT AS WE HAD BEEN GIVEN NO INDICATION THAT TWR HAD BEEN OPEN; UNTIL THAT MOMENT. WE RESPONDED BY SAYING THAT WE DID NOT KNOW THAT TWR WAS IN OP SINCE ATIS SAID THE TWR WAS CLOSED. IN HINDSIGHT; THIS COULD HAVE BEEN PREVENTED BY CHKING THE ARPT FACILITY DIRECTORY; OR THE ATIS THAT WAS BEING BROADCAST; FOR THE TIME THAT THE TWR SHOULD HAVE BEEN OPEN. WE COULD HAVE QUERIED ANYONE ON THE CTAF FREQ TO ASCERTAIN THE STATUS OF THE TWR OR; WE COULD HAVE SIMPLY MADE A TA THAT WE WERE TAXIING TO THE ACTIVE AND THAT MAY HAVE ALERTED ATC. IN ADDITION; ATC PERSONNEL MADE NO ATTEMPT TO CONTACT US ON ANY FREQ BEFORE HAND AND APPARENTLY DID NOT LOOK OUTSIDE THEIR WINDOW AT ANY TIME BEFORE OUR CALL.