# In a metric space X, if A is connected, is its interior connected?

I believe that int(A) is also connected. I tried to use an argument by contradiction, that is to say I supposed that int(A) is not conneted, but without success

• umm It's interior is the empty set Commented Jan 9, 2017 at 5:47
• @MyGlasses Sets that has empty interior would not be any good example for this as the empty set is connected (as any set with no more than one element trivially is). Such sets could only be used as example of the opposite: non-connected sets with connected interior (for example $\{0,1\}$ which is non-connected, but has connected interior). Commented Jan 9, 2017 at 7:21
• Consider the union of first and third quadrants on the plane along with the axes. Commented Jan 9, 2017 at 7:34
• Note that it is a fairly reasonable convention that the empty set is not connected (it has zero connected components, not one). Of course this is not an ideal example, since it immediately raises the question of whether requiring the interior to be nonempty changes the answer. (No, as the comment above and the answer below show.) Commented Jan 9, 2017 at 14:04
• For $n \ge 2$, you can take any two connected sets in $\mathbb R^n$ with non-empty interiors, whose union is not connected, and join them by a straight line. Commented Jan 9, 2017 at 20:31

Consider $X=\mathbb R^2$ and $$A=([-2,0]\times[-2,0])\cup([0,2]\times[0,2])$$which is connected, while $\text{int}(A)$ is not connected.
To see this consider the continuous function $f:\mathbb R^2\to\mathbb R$ is defined by $f(x,y)=x+y$. Let $U=f^{-1}(0,+\infty)$ which is open in $\mathbb R^2$ and so $U\cap\text{int}(A)$ is open in $\text{int}(A)$. Also, since $(0,0)\notin\text{int}(A)$, so for all $(x,y)\in\text{int}(A)$, $f(x,y)\neq0$ and $U\cap\text{int}(A)=f^{-1}[0,+\infty)\cap\text{int}(A)$ is closed in $\text{int}(A)$. Furthermore, $(1,1)=f^{-1}(2)\in U\cap\text{int}(A)$ shows that $U\cap\text{int}(A)\neq\emptyset$ while $(-1,-1)\in\text{int}(A)$ and $(-1,-1)\notin U$ shows that $U\cap\text{int}(A)\neq\text{int}(A)$.