This happened to me as well,
I had actually brought the subject up the first time I saw her, given the location we had met and the fact i would occasionally be in the general area she worked.....we agreed that discretion, for both our sakes, is best maintained at all times.
Sure enough, while out for shopping trip, we run into each other, and with a polite smile and a gentle nod of our heads, we politely acknowledged each other.....and moved on without stopping or conversing.
I sent an email afterwards, wishing her a great day and she was appreciative that i did not stop to chat, as was I she didn't....you never know who is lurking nearby, family, friend...business partner....
Discretion is always best maintained in my opinion....