"Close to" and "near" have the same basic meaning.
Are these sentences correct:
1) He ran up close to her.
2) He ran up near her.
Do these mean:
a) He ran up to her. (He approached her, running.)
b) He was running not far away from her. (He was running and he was close to her.)
They both have the meaning of a), 'to approach'.
Wear short sleeves! Support your right to bare arms!