Dense set equivalent definitions

[HAT]

Hello all, I am an undergraduate student who is studying real analysis from Rudin's POMA and I am trying to prove that these two definitions that I have for dense sets are equivalent:
1) Given a metric space X and E ⊂ X ; E is dense in X iff every point of X is a limit point of E or E = X or both of these are true.
2) Given a metric space X and E ⊂ X; E is dense in X iff the intersection of E and every non-empty open set of X is non-empty.
In an attempt to prove the equivalence I have encountered an example which I can't get my head around it.
Given the set X such that X consists of all points $s_n$ , where $s_n = \sum_{k=0}^n (1/2)^n$ for all n ≥ 0, and 2 as well. Now define the metric for such a set to be the same as that of ℝ. Then X is a metric space. Now according to definition (1) the only dense set in X is X itself, but according to (2) the set V = X - {2} is a dense set in X besides X as well. However we should not have such a problem. So could you please point out what I am doing wrong. Thank you.

 

[fresh_42]

I prefer the definition, $E$ is dense, iff $\bar{E}=X$, which is definition one.

I had difficulties to correct definition two, as it is a bit of a sloppy notation for limit points. Maybe it's better to write it with open neighborhoods $U_x$ of a point $x$ and require $E \cap (U_x-\{x\}) \neq \emptyset$, but that's basically definition one.

 

[member 587159]

I prefer the definition, $E$ is dense, iff $\bar{E}=X$, which is definition one.

I had difficulties to correct definition two, as it is a bit of a sloppy notation for limit points. Maybe it's better to write it with open neighborhoods $U_x$ of a point $x$ and require $E \cap (U_x-\{x\}) \neq \emptyset$, but that's basically definition one.

You mean basically definition 2.

 

[fresh_42]

Have I confused something? I'm used to define points of a set as automatically limit points. That's why it is better to define $E\subseteq X$ is dense, if $\bar{E} =X$, which avoids this special case. So let's see, what we get for $E=X-\{2\}$. With my definition it is dense in $X$.

Now if $\{1\} \in E$ is no limit point, because without $x=1$ there are no small non-empty open sets around it, then according to definition 1) $E$ wasn't dense, which is wrong.

On the other hand, in definition 2) we have $E \cap \{1\} = \{1\}$ as an intersection of $E$ with an open non-empty set which is non-empty, so the points of $E$ are included. This is correct.

So, yes, you are right, I confused the two. The second definition is right and the first one is not. Thanks for the correction.

 

[mathwonk]

This may not be of interest, since I don't know why you are using POMA, but when I hear that, my first reaction is to suggest you read a different book, like anything by George Simmons or Sterling Berberian, if you want a book tnat teaches you something in a user friendly way. Or if you stick with Rudin but have difficulty understanding at least remember it isn't necessarily your fault. I myself learned metric spaces from lectures by George Mackey and never needed to read any book on it afterwards, since they were so clear. He wrote a book on complex variables with an introductory chapter on metric spaces and elementary topology that might be useful. Also Dieudonne' has a great book on Foundations of modern analysis with a chapter on metric spaces but he is not easy reading. In my opinion your choice of book is making things harder unnecessarily, but anyway good luck. It is of course a standard and well respected by many professional analysts.

 

[fresh_42]

... my first reaction is to suggest you read a different book ...

I recently came upon this free topology book on AMS OpenMathNotes, which looked very promising at first sight, especially with respect to analysis: https://www.ams.org/open-math-notes/omn-view-listing?listingId=110653