No sure what definition of ‘open’ you are using. This one is standard: A setSis open iff for each x isSthere is a open setO_xwhich contains x and is a subset ofS.

Do you see how part A) results directly from that definition.

For part B), any open set ofSis a subset ofS^o.

For part C), a point t is in int(S) iff there is a open setO_twhich contains t and is a subset ofS.