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Let $(X,d)$ be a metric space and $a_1,a_2,\ldots\in X$. Define $A=\{a_n:n\in\mathbf N\}$. Is $A$ closed in $(X,d)$?

And is $A$ closed when X is a topological space?

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5 Answers 5

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Another example of a countable, non-closed set is $\mathbb{Q}$ inside of $\mathbb{R}$ with the usual topology, since $\mathbb{Q}$ is dense and there are irrational numbers in $\mathbb{R}$.

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No. Consider $a_n = \frac{1}{n}$ and $X= \Bbb{R}$ equipped with the euclidean metric. The set $A$ is not closed because $0$ is a limit point of $A$ such that for no $n$ is $a_n$ ever equal to $0$.

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    $\begingroup$ So the negative answer is due to possible cluster points of $(a_n)_n$ not in $A$. $\endgroup$
    – Seirios
    Jul 9, 2012 at 12:55
  • $\begingroup$ @Seirios: Yes, but I would write it cluster points of $a_n$, with just one subscript. $\endgroup$ Jul 9, 2012 at 13:12
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    $\begingroup$ That is what "not closed" means: there are cluster points not in the set. $\endgroup$
    – GEdgar
    Jul 9, 2012 at 14:13
  • $\begingroup$ I disagree with your second example. Sequence $\{x^n\}$ is closed in $C[0,1]$ equipped with the sup metric. $\endgroup$
    – GEdgar
    Jul 9, 2012 at 14:15
  • $\begingroup$ @GEdgar I forgot that the pointwise limit of $\{x^n\}$ lies outside of $C[0,1]$. $\endgroup$
    – user38268
    Jul 10, 2012 at 8:40
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Let $X$ be an infinite topological space. Then:

1) If $X$ is discrete, then (all subsets are closed, so) every countably infinite subset is closed.

2) If $X$ is non-discrete, separated and first-countable -- in particular, if $X$ is non-discrete and metrizable -- then there is a non-isolated point $x$ and a sequence $\{x_n\}_{n=1}^{\infty}$ of distinct elements of $X \setminus \{x\}$ converging to $x$. Thus $A_1 = \{ x_n: n \in \mathbb{Z}^+ \}$ is a countably infinite subset which is not closed, and $A_2 = A_1 \cup \{x\}$ is a countably infinite subset which is closed.

3) Above the hypothesis of first countability can be somewhat weakened: it is enough to assume instead that $X$ is sequential. But it cannot be dropped entirely: let $X$ be an uncountable set endowed with the cofinite topology, in which a proper subset is closed if and only if it is finite. Then $X$ is a separated space which has no countably infinite closed subsets.

At the moment I am not seeing a clean necessary and sufficient condition on a general topological space for it to admit a countably infinite closed subset. Maybe someone else can do better?

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Here's an example from functional analysis:

Consider $B[0,1] = \{f: [0,1] \to \mathbb R \mid f \text{ bounded }\}$ with the metric $d(f,g) = \int_{[0,1]} |f(x)-g(x)| dx$. Then we define the following countable subset: $S = \{ f_n (x) \mid n \in \mathbb N_{\geq 2}\} \subset C[0,1] \subset B[0,1] $ where

$$ f_n(x) = \begin{cases} 0 & x \in [0, \frac{1}{2}-\frac1n] \\ nx - \frac{n}{2} + 1 & x \in [\frac{1}{2}-\frac1n, \frac{1}{2}]\\ 1 & x \in [\frac12, 1] \end{cases}$$

Then $f_n$ is continuous. Its pointwise limit $f(x)$, which is also its limit with respect to $d(\cdot, \cdot)$, is the (discontinuous) function that is $0$ on $[0,\frac12)$ and $1$ on $[\frac12,1]$ so the limit of $f_n$ is not in $C[0,1]$ and hence also cannot be in $S$.

Note:

If you use a different metric, such as for example $d(x,y) = \sup_{x \in [0,1]} |f(x) - g(x)|$, the pointwise limit $f(x)$ will no longer be a limit of $f_n$ with respect to the metric because the sequence is no longer Cauchy.

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  • $\begingroup$ @BenjaLim Did you miss a "not" in that comment? (we want a set that is not closed) $\endgroup$ Jul 10, 2012 at 10:52
  • $\begingroup$ Sorry, I meant to say how does this show that family $f_n$ is not closed? I think the family $f_n$ has no limit points (as you noted the pointwise limit is not in $C[0,1]$ so we can't count it as a limit point in the space you're considering). $\endgroup$
    – user38268
    Jul 10, 2012 at 10:55
  • $\begingroup$ @BenjaLim Oh dear. Well spotted. I can salvage the example though I think. $\endgroup$ Jul 10, 2012 at 11:02
  • $\begingroup$ @BenjaLim Look, I think I fixed it. : ) $\endgroup$ Jul 10, 2012 at 11:04
  • $\begingroup$ @BenjaLim I think it's clear as it is : ) $\endgroup$ Jul 10, 2012 at 11:41
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The definition of countably compactness maybe helpful for your question:

$X$ is countably compact, i.e., for every infinite subset $A$ there exists a cluster $a$ of A.

(Another equivalent definition of countably compactness is this: for every countable open cover $U$ of $X$, there exists finite subcover $V \subset U$ which covers the space $X$.)

So, if $a \in A$, then $A$ is closed in $X$; and if your $A$ does not contain his cluster point, then it is not closed.

Remark: However, metric space or topological space does not imply countably compactness. it is another topological property. But we have countably compactness = compactness in the metric spaces.

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