What is compassion?

I think most people try to be compassionate when someone they know is going through something rough.  Sometimes it is hard to know how to convey that though.  Case in point - an acquaintance of mine lost her husband 2 weeks ago.  He passed away suddenly at the age of 46.  He hadn't been through any long illness, had any type of injury, no signs of any problems.  This acquaintance is someone who goes to the same boot camp class I go to at the gym.  We are friendly at class but our paths don't cross outside of the gym, and even though we often have social gatherings for the group, she & I have never been at one together.  Another person in the class is good friends with her and they suggested she would need support down the road when the family is gone, etc.  Since she wasn't someone I would call a friend, I did not attend any of the services.  I have not spoken with her.    Then, she showed up at class Saturday in the middle of class.  Towards the end of class we happened to be side by side and a little isolated from the rest of the class.  Normally in that situation I would chit chat with whomever was there but I didn't feel comfortable making idle chatter without giving condolences.  But then I felt strange offering condolences mid-workout because I felt like her coming to class was a way of trying to get back to her day to day activities and find a new normal.  So I'm torn but finally I ask how she is and tell her I'm sorry and that she's been on my mind.  She got really choked up and I found myself apologizing for bringing it up.  She reassured me that it was ok and that it's good to know people care.   I felt terrible, but I really didn't know how to handle being there with her and not having spoken to her earlier.  (Not that it's about how I feel, because I know, it's not.)  So in a situation like this, is is more compassionate to ignore the situation and make idle chit chat, or is it more compassionate to extend the sympathy, even if it's not the best place and time?