Is my understanding of open sets and bases in topology correct?

[Question Man]

My brain is giving me confusions.
Which of these is true?
1) Given a topology T and basis B, a set U is open iff for every x in U there exists basis element B with x belonging to B, and B contained in U.
2) Given a topology T and basis B, a set U is open iff for every x in U there exists open set V with x belonging to V, and V contained in U.
3) Given a topology T and basis B, a set U is open iff exery point of U belongs to the closure of U.

 

[Crosstalk]

1) Given a topology T and basis B, a set U is open iff for every x in U there exists basis element B with x belonging to B, and B contained in U.

The definition of open set that I was taught in topology* is that a set is open if it is the union of neighborhoods. How is this similar to your statement?

2) Given a topology T and basis B, a set U is open iff for every x in U there exists open set V with x belonging to V, and V contained in U.

This can most easily be answered by relating it to the "union of neighborhoods" definition, as a union of unions of neighborhoods is a union of neighborhoods. Thus an open set is a union of open sets.

3) Given a topology T and basis B, a set U is open iff every point of U belongs to the closure of U.

The definition I have learned for closure* is that the closure of a set is the set of all points near to the set. Is an element of the set near to that set?

* I worded it this way in case different topology courses teach different (but equivalent) definitions for open sets.

EDIT: Changed to follow homework help guidelines.

 

[Rasalhague]

1. "Given a topology T and basis B, a set U is open iff for every x in U there exists basis element b in B with x in b, and b a subset of U."

For the "if", consider the definition of basis. For the "only if", recall that the elements of B are open, and consider what the definition of topological space says about unions of open sets.