INSTRUCTOR PLT WITH PVT PLT STUDENT LANDED AND TOOK OFF AND RELANDED AT A CLOSED ARPT DURING SHORT FIELD TKOF AND LNDG PRACTICE.

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I TOOK A PVT PLT UP FOR SOME PRACTICE SHORT FIELD TKOF AND LNDGS TO PLUMB ISLAND ARPT; A GREAT ARPT FOR THESE LNDGS AND TKOFS (2500 FT RWY). I BROUGHT HIM TO THE TFC PATTERN FOR L DOWNWIND FOR RWY 28. I CALLED ON UNICOM 123.0 TO LET OTHER TFC KNOW OF OUR INTENTIONS. AS WE CAME IN ON FINAL THE PVT PLT WAS LOW AND EXPLAINED I WANTED HIM TO CLR A 100 FT OBSTACLE JUST OFF THE RWY. AFTER WE LANDED WE CALLED TFC AND SAID WE WERE BACK TAXIING TO RWY 28. AS WE GOT TO THE END OF RWY 28 I NOTICED AN X ON THAT END OF THE RWY. BEFORE I LEFT FOR THE FLT I CHKED THE WX AND NOTAMS AND THERE WAS NOTHING ABOUT PLUMB ISLAND BEING CLOSED. I CHKED AGAIN (MONDAY) AND STILL NO NOTAMS ON ARPT CLOSING. ANYWAY I THOUGHT MAYBE THEY WERE GOING TO BE WORKING ON RWY 28; THAT'S WHY THERE WAS AN X. WE DID A SHORT FIELD TKOF; CLBED OUT AND CAME BACK IN FOR RWY 10 WHERE THERE ARE TREES AT THE END OF RWY FOR A BETTER EFFECT OF SHORT FIELD LNDG (WINDS LIGHT). WE CAME IN AND LANDED I NOTICED THE X ON RWY 10. AFTER WE LANDED WE TOOK OFF AND I LOOKED BACK AT RWY AND THEN NOTICED AN X ON THE GRASS STRIP ALSO AND THEN REALLY REALIZED ARPT WAS CLOSED. BECAUSE I WAS EXPLAINING THE MANEUVER AND MAKING SURE HE WAS CONFIGURED FOR THE APCH I DID NOT SEE THE X. THIS WAS MY ERROR; BUT I HONESTLY DID NOT MEAN TO LAND AT AN ARPT THAT WAS CLOSED. I DID NOT HEAR ANYTHING ABOUT THE ARPT BEING CLOSED AND THERE WERE NO NOTAMS ABOUT IT BEING CLOSED. I APOLOGIZE FOR MY ERROR AND HOPE THIS DOES NOT EFFECT MY AVIATION CAREER AS I TRULY BELIEVE I AM A SAFE PLT AND FOLLOW FAA REGS.