Interior of Natural Numbers in a metric space

Question

I'm trying to understand the definition of open sets and interior points in a metric space.

I'm not sure why $$ Int(\mathbb{N})=\phi.$$

$ A\subseteq X, a \in X $, then $a$ is said to be an Interior Point of $A$ if $ \exists r \in \mathbb{R} >0 $ such that $ U(a,r) \subseteq A. $

Where $U(a,r)$ denotes the open ball center $a$ radius $r$.

Now applying this definition letting $X= \mathbb{R}$ and $A=\mathbb{N}$ and the usual metric on $\mathbb{R}$, then could I not take $a=2, r=3$ then take say $x=1$, then $d(a,x)=d(2,1)=|2-1|= 1 < 3.$ Clearly $1 \in \mathbb{N}$...

Or have I misunderstood and it's the case that every point in the open ball has to belong to $A$?

Thanks.

All the definitions I've looked at just have it as $\exists r >0$ but does that still mean $\forall r$? — Numbertheorylearner, Mar 20 '17 at 18:43
@Giulio: That doesn't look like the correct definition. With your definion, $a=1$ would not be an interior point of $A=[0,2]$, because $r=42$ does not satisfy $U(1,42)\subseteq [0,2]$. — hmakholm left over Monica, Mar 20 '17 at 18:44
I got my original definition from this video(44:49): https://www.youtube.com/watch?v=Vfs1-DYywL4&index=16&list=PLbMVogVj5nJSxFihV-ec4A3z_FOGPRCo- — Numbertheorylearner, Mar 20 '17 at 18:46

score 3 · Accepted Answer · answered Mar 20 '17 at 18:46

3

No, your reasoning doesn't work.

If, as you say, you take $a=2, r=3$ you have to ask whether $U(2,3)\subseteq \mathbb N$.

But $U(2,3)$ is the open interval $(-1,5)$, which is not a subset of $\mathbb N$. For example, $(-1,5)$ contains the point $\frac12$ which is not in $\mathbb N$.

answered Mar 20 '17 at 18:46

hmakholm left over Monica

286,031

1

So then am I right in thinking that it's all points inside the open ball have to lie inside A? – Numbertheorylearner Mar 20 '17 at 18:48
@Numbertheorylearner: Yes, that is the meaning of the symbol $\subseteq$ in "$U(a,r)\subseteq A$". – hmakholm left over Monica Mar 20 '17 at 18:49
Epic failure moment! Thanks. – Numbertheorylearner Mar 20 '17 at 18:53

Interior of Natural Numbers in a metric space

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