"Persephone's my own place, a little ways from here. Maybe a ten minute walk or so, at the most."

Although walking from point A to point B could pretty quickly turn into a contact sport if you don't stay aware of your surroundings. Sinclair thinks that maybe it's getting to be time to start carrying a pistol, and suddenly feels somewhat relieved that Jack is armed. Not that Jack would necessarily be inclined to save his life, should something happen, but worst comes to worst Sinclair believes he might be able to get Jack's gun from him. Somehow.

"Now, I don't want you to be startled when we get there, son," he continues. "Persephone's a detention center for Rapture. My office isn't anywhere near all that, though, you don't have to worry."