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I am currently reading the book A combinatorial introduction to topology by Michael Henle. Under "Compactness and Connectedness" there is the following definition which I didn't understand at all. I do know the definition of nearness.

Let $P=\{P_1,P_2,P_3,\ldots \}$ be a sequence of points. The point $Q$ is near the sequence $P=\{P_n\}$ if either $Q=P_n$ for an infinite number of terms of the sequence or $Q=P_n$ only a finite number of times and $Q$ is near the set of other values of $P$.

Any help would be greatly appreciated.

Alexander Gruber
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RagingBull
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  • Are you sure you typed it in right? – Alexander Gruber Jan 19 '13 at 23:38
  • In my semi-professional opinion, this definition makes absolutely no sense. (I also found the book and this definition, and it is copied down correctly.) I have no idea what they meant. – Dylan Wilson Jan 19 '13 at 23:45
  • @AlexanderGruber-yea i typed it right...as dylan said even he has the book and the definition has been copied correctly – RagingBull Jan 20 '13 at 00:00
  • @DylanWilson-as you can see in the book,the author has used this definition to define continuity.Can you explain what does nearness to a sequence mean? – RagingBull Jan 20 '13 at 00:07
  • i came across this but the answer here seems very clumsy http://mathforum.org/kb/thread.jspa?forumID=13&threadID=40520&messageID=138008 – RagingBull Jan 20 '13 at 00:09
  • Page 13 of the book gives the definition of "$Q$ is near the set $A$." – Andrés E. Caicedo Jan 20 '13 at 00:17
  • I know the meaning of nearness to sets but the author seems to have made a distinction between the definition of "nearness to sets" and "nearness to sequences".Had the two been the same the author wouldn't have written the above definition.Can you explain what does this definition mean?What does Q=Pn mean? – RagingBull Jan 20 '13 at 00:19
  • @AndresCaicedo, please, could you write the definition of $Q$ is near the set $A$? – Sigur Jan 20 '13 at 00:33
  • How does the book define Q is near a set A? Of course, you can regard a sequence as a set--it's just the range of the function from the positive integers $f(n)=p_n$. – Josh Jan 20 '13 at 00:34

1 Answers1

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The definition of "$Q$ is near the set $A$" is "every neighborhood of $P$ contains a point of $A$".

Now, a sequence is really a function $P$ from the set of positive integers to the space. We write $P_n$ instead of $P(n)$, but $P$ is really a function. The author uses set notation, and writes $P=\{P_1,P_2,\dots\}$, but this is not quite correct, since there is a difference between the sequence, which is a function, and its range, which is a set. So, for example, if $P$ is the sequence $P_1=1,P_2=-1,P_3=1,P_4=-1,\dots$ and if $P'$ is the sequence $P'_1=P'_2=1,P'_3=P'_4=P'_5=\dots=-1$, then both are very different as sequences, but have the same range, namely $\{-1,1\}$.

To say that $Q$ is near a sequence $P$ means that, either for infinitely many values of $n$, $Q$ is the value taken by the function $P$ at $n$, $Q=P_n$, or else this fails, but $Q$ is near the set resulting from excluding $Q$ from the range of $P$.

To illustrate, for the sequence $P$ given by $P_1=1,P_2=-1,P_3=1,P_4=-1,\dots$ Here, the point $Q=1$ is near $P$, because $Q=P_1=P_3=P_5=\dots$ Also, the point $R=-1$ is near $P$, because $R=P_2=P_4=\dots$ On the other hand, no other point $S$ is near $P$, because if $S\ne1$ and $S\ne-1$, then there is a neighborhood of $S$ so small that neither $1$ nor $-1$ is in it.

Now, look at the sequence $P'$ given by $P'_1=P'_2=1,P'_3=P'_4=P'_5=\dots=-1$. As before, if $S\ne1$ and $S\ne -1$, then $S$ is not near $P'$. Also, $R=-1$ is near $P'$ as $R=P'_3=P'_4=P'_5=\dots$ However, $Q=1$ is not near $P'$, because there are only two values of $n$ for which $Q=P'_n$, namely, $n=1$ or $n=2$. The set of other values of $P'$ is just $-1$, and there is a neighborhood of $Q$ that misses $-1$, so $Q$ is not near the set of other values.

All this being said, in practice, most sequences $P$ we are interested in will not repeat values, so a point $Q$ will be near $P$ iff $Q$ is not in the range of $P$, but $Q$ is near the range of $P$. For example, $0$ is near the sequence $P$ given by $P_n=1/n$ for $n=1,2,\dots$

It is in order to avoid having to phrase arguments awkwardly, and to avoid unnecessarily splitting them into two cases, that the definition is presented to cover both possibilities, when $Q$ is repeatedly listed in the sequence $P$, and when is actually being ``approached'' by the sequence.

Andrés E. Caicedo
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  • @andres-thanks a lot sir...very good explanation indeed – RagingBull Jan 20 '13 at 00:58
  • @Andres-Why such a distinction between the definition of nearness to set and nearness to sequence?A Sequence is a special kind of set.Consider the sequence Pn=1/n,n=1,2,. Let Q=1/3.If we go by the definition then Q=Pn for n=3 but it is not near the set of other values of P.But if we consider the sequence as a set then Q=1/3 is near the set P because every neighborhood of Q contains a point of P which is Q itself.I think it is correct to say that any point in a set A is near the set A.Am I making a mistake somewhere?The problem seems to arise when we consider a point from the range of Pn. – RagingBull Jan 20 '13 at 14:29
  • @BasantSharma This is true. The notion of nearness is useful when defining the closure of a set; since we want the closure to contain the set, it is natural to say that points in the set are near the set. But the notion is also useful to define when a sequence converges, and the latter should be a notion robust enough that does not change if we replace the sequence with one of its tails. The definition is designed so that this happens, and in fact, a point is near a sequence $P$ precisely when there is a subsequence of $P$ that converges to the point. – Andrés E. Caicedo Jan 20 '13 at 15:50
  • @BasantSharma (There is of course a relation. For example, in metric spaces, a point is in the closure of a set if and only if it is the limit of a sequence of points in the set.) – Andrés E. Caicedo Jan 20 '13 at 15:50