Let X be a topological space and let A be a subset of X

Question

Let $X$ be a topological space and let $A$ be a subset of $X$. Which of the following statements are true?

a. If $A$ is dense in $X$, then $A^o$ (the interior of A), is also dense in $X$.

b. If $A$ is dense in $X$, then $X\setminus A$ is nowhere dense.

c. If $A$ is nowhere dense, then $X\setminus A$ is dense.

Attempt a.False Try out $X=\mathbb{R}$ and $A=\mathbb{R}$

b,Flase Try out $X=\mathbb{R}$ and $A=\mathbb{Q}$

c.I Think its gonna be true because if $X=\mathbb{R}$ and $A=\mathbb{N}$ but hiw can I prove

P.S. I am new and poor in Latex. So forgive any mistakes commited.

Answer 1

a. It is false, but not because of your example. Take $X=\mathbb R$ and $A=\mathbb Q$.

b. It is false. Again, take $X=\mathbb R$ and $A=\mathbb Q$.

c. It is true, but an example doesn't prove it. Suppose that $X\setminus A$ is not dense. Let $B=X\setminus\overline{X\setminus A}$. Then $B$ is a non-empty open set and $B\subset A$. Therefore, $B\subset\mathring{\overline A}$, which is impossible, since $\mathring{\overline A}=\emptyset$.

Answer 2

Jose's answer is correct and 100% complete. I just want to say that for (c), you can also use the fact that $$\overline{A^{c}}=(int(A))^c$$ Having that in mind we can easily conclude that if $A$ is nowhere dense , meaning $int(\overline{A})=\emptyset$ then :

$$\overline{A^c}=(int(A))^c\supset(int(\overline{A}))^c=(\emptyset)^c=X$$