1. ## 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!

2. 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 $\displaystyle C$ is a closed set containing $\displaystyle E$ then $\displaystyle x\in C$ where $\displaystyle x$ is a boundary point of $\displaystyle E$.

If $\displaystyle x\in (\text{int}(E) \cup \text{int}(E^*))^*$ then $\displaystyle x\in ( \text{int}(E)^* \cap \text{int}(E^*)^*)$.
Thus, $\displaystyle x\in \text{int}(E)^*$ and $\displaystyle x\in \text{int}(E^*)^*$.

Note $\displaystyle E\subseteq C \implies C^* \subseteq E^* \implies \text{int}(C^*) \subseteq \text{int}(E^*)$.

Since $\displaystyle C$ is closed it means $\displaystyle C*$ is open therefore $\displaystyle \text{int}(C*) = C^*$.

Therefore, $\displaystyle C^* \subseteq \text{int}(E^*) \implies \text{int}(E^*)^* \subseteq (C^*)^* = C$

It follows that $\displaystyle x\in \text{int}(E^*)^* \subseteq C$

3. 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?