Thread: some topology questions about closure and boundary

so i was reviewing topology and i came across some questions that i was not able to prove:

1)show that for a set E in R^n, the boundary of E is contained in the closure of E.
the given definition is that x is a boundary point of E if x is in the complement of int(E) U int(complement of E), and that the closure of E is the intersection of all closed sets containing E. i tried picking an x in the boundary of E, but couldnt relate it to the closure in anyway through set theoretic methods.

2) show that for a set E in R^n, closure of E = E U (the boundary of E).
again i know that the way to do it is set inclusion, but still have trouble relating the boundary of E to the closure of E.

thanks!

Originally Posted by squarerootof2

1)show that for a set E in R^n, the boundary of E is contained in the closure of E.
the given definition is that x is a boundary point of E if x is in the complement of int(E) U int(complement of E), and that the closure of E is the intersection of all closed sets containing E. i tried picking an x in the boundary of E, but couldnt relate it to the closure in anyway through set theoretic methods.

It is sufficient to show that if is a closed set containing then where is a boundary point of .

If then .
Thus, and .

Note .

Since is closed it means is open therefore .

Therefore,

It follows that

i was thinking about it and i think i found a better version of the proofs. can we just say boundary of E is contained in boundary of the closure of E, which should be contained in the closure of E as E is a closed set?