PLT IN C172 ADVISES BKL CLRNC DELIVERY THAT HE WANTED A 'VFR DEP' ON A HDG OF 240 DEGS. AFTER TKOF; THE TWR ADVISED THE PLT TO TURN TO HDG 240 DEGS. THE PLT REMAINED ON THE HDG AND WONDERED WHY THE CTLR DIDN'T ADVISE HIM TO CHANGE FREQ.

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ON SEP/XA/98 AT APPROX XA15; I CONTACTED BKL CLRNC DELIVERY ON 121.90 AND INFORMED THE CTLR THAT I WAS A CESSNA SKYHAWK 172/U AND REQUESTED A VFR DEP ON A HDG OF 240 DEGS WHILE CLBING TO 6500 FT MSL. CLRNC DELIVERY ACKNOWLEDGED MY REQUEST AND TOLD ME TO SQUAWK 1200. I THEN CONTACTED GND CTL ON 121.90 (SAME AS CLRNC) AND REQUESTED PERMISSION TO TAXI TO THE ACTIVE RWY FOR RUNUP AND TKOF. GND CTL ADVISED ME TO TAXI TO AND HOLD SHORT OF RWY 6L FOR MY RUNUP AND TO CONTACT THE TWR ON 124.30. AFTER MY RUNUP I CONTACTED THE TWR; TOLD THE CTLR MY RUNUP WAS COMPLETE AND THAT I WAS READY FOR TKOF ON RWY 6L. THE TWR CLRED ME FOR TKOF AND ADVISED ME TO TURN TO A HDG OF 240 DEGS AND TO BE ADVISED OF HELI TFC OVER THE DOWNTOWN AREA AT 1300 FT. I ACKNOWLEDGED THE TURN TO A HDG OF 240 DEGS AS WELL AS THE TFC OVER THE DOWNTOWN AREA. AFTER I TOOK OFF; I INITIATED MY TURN TO A HDG OF 240 DEGS; AS I WAS CLBING BTWN 3000-4000 FT ON HDG 240 DEGS. I NOTICED THAT I WAS WITHIN 5 NM OF CLE ARPT. BURKE LAKEFRONT TWR AT NO TIME ADVISED ME TO CONTACT ZOB AND DID NOT ADVISE ME OF ANY FREQ CHANGE. I REMAINED ON MY HDG OF 240 DEGS AS WELL AS BURKE LAKEFRONT'S TWR FREQ UNTIL I WAS 20 NM N OF CMH. AT THIS TIME I CONTACTED CMH APCH AND INFORMED THEM THAT I WAS INBOUND FOR THE FULL STOP AT TZR UNDER VFR AT 6500 FT. CMH APCH REQUESTED THAT I STAY ABOVE 3000 FT AND TO SQUAWK XXXX. AFTER SQUAWKING XXXX AND IDENTING MY XPONDER; CMH APCH ACKNOWLEDGED RADAR CONTACT AND ADVISED ME TO FLY DIRECT TO BOLTON FIELD. AT APPROX 8.5 NM FROM BOLTON; I CONTACTED CMH CTL TO INFORM THEM THAT I WAS STARTING MY DSCNT TO PATTERN ALT FOR THE FULL STOP AT BOLTON. CMH CTL ACKNOWLEDGED MY REQUEST AND ADVISED ME TO DSND AT MY DISCRETION AND TO CONTACT BOLTON TWR ON 128.10. I CONTACTED BOLTON TWR; THEY CLRED ME FOR LNDG ON RWY 22; NOT ABLE TO MAKE THE LNDG DUE TO BEING 1000 FT MSL ABOVE RWY 22'S THRESHOLD. BOLTON TWR CLRED ME FOR THE GAR AND TO LAND BEHIND THE PATTERN TFC. I FINALLY LANDED AT BOLTON AND TAXIED TO THE PARKING AREA WITH BOLTON GND CTL'S PERMISSION. MY CONCERN IS THAT BURKE LAKEFRONT AT NO TIME ADVISED ME TO CONTACT A ZOB CTLR WHILE ENTERING OR BEING WITHIN CLE'S CLASS B AIRSPACE. AFTER DISCUSSING THIS SIT WITH ANOTHER PLT; HE INFORMED THAT IT MIGHT HAVE BEEN MY RESPONSIBILITY TO CONTACT THE ZOB CTLR. THIS WAS MY FIRST FLT FROM THE CLE AREA. ACCUSTOMED TO BEING HANDED OFF TO A DEP FREQ FROM THE TWR FREQ AT CMH; I REMAINED ON BURKE LAKEFRONT'S TWR FREQ UNTIL I WAS WELL CLR OF CLE'S CLASS B AIRSPACE. BURKE LAKEFRONT'S CLRNC DELIVERY OR TWR CTLR COULD HAVE ADVISED ME TO CONTACT CLE CTL AFTER TKOF SINCE THEY WERE WELL AWARE THAT MY HDG WAS GOING TO CUT INTO THE CLASS B AIRSPACE.