My mother has little red sores on her ankles and legs. I've used moistureizer and topic antibiotics, but it doesn't help for long. How can these be treated?

It's important that she see a doctor. These sores may be an indication of an underlying problem. It's one thing to have a small sore and treat it topically - if it goes away, there's generally no problem. These keep coming back, so there's a reason that isn't apparent. Please have a doctor exam her.
Take care,
Carol

Could they be bites? If they are on her ankles and legs they might be flea bites. Do have her checked by a Doctor and if they are flea bites you will need to find the source of the problem and have an Exterminator come in.

Could they be bedbug bites?

You should definitely have a dermatologist take a look at the bumps. A good rule is that any symptom that isn't life threatening, but a nusiance should be evaluated if it lasts more than 2 weeks.

That's the first thing I thought of too, was flea bites.