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Random Talk on Random Thoughts

Closedness vs Completeness

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To prove a limit, we need to find a number before. The definition gives us no hints on how to find it. Cauchy’s Criterion for convergence is a good method for finding it numerically in “certain spaces” (e.g. $\R,L^p,L^\infty$) because the definition is intrinsic.

Unluckily, there’re some “strange spaces” where Cauchy’s Criterion isn’t enough to give a convergence. The simplest one is $\Q$ equipped with the usual metric in $\R$.

One knows that an incomplete metric space $X$ can extended to form a complete one $X^*$ (which is unique up to isometric isomorphism) so that $\iota(X)$ is dense in $X^*$. That is, $\overline{\iota(X)} = X^*$.


I confused $\overline{X}$ with $X^*$. In other words, what’s the difference between the completion of $X$ and the closure of $X$?


General answer

To see this, we note that the closedness of a set is discussed relative to another set, so if one stays inside the incomplete space $X$, then one notices that $X$ is closed in $X$. However, even though one doesn’t know about $X^*$, one still realises that $X$ is incomplete.

If one becomes wiser and seeks to “complete a given Cauchy sequence” (i.e. find a limit for a Cauchy sequence in $X$), it’s impossible to live inside $X$—one needs to escape from $X$ to $X^*$ so as to enjoy more freedom.

Claim: $X$ isn’t closed in $X^*$.

Approach: We won’t start from the topological definition (open complement). Instead, we’ll argue with sequences.

Proof: Since $X$ is incomplete, there exists a Cauchy sequence $(x_n)$ in $X$ so that it has no limit in $X$. The completeness of $X^*$ is applied to construct a limit $x$ in $X^*$. If one allows $X$ to be closed, then $x$ will stay inside $X$, which contradicts the fact that $(x_n)$ has no limit in $X$.

Even though living in $X$ is sad, and it seems to be hard to escape (because $X$ is closed in $X$), if one has a chance to travel outside, one will see that $X$ is not really closed—it is possible to find a sequence $(x_n)$ to escape from $X$.

Specific example

$\Q$ is closed in $\Q$, but it isn’t closed in $\R$. Therefore, the concept of closedness depends on the bigger space. However, the concept of completeness only depends on $\Q$ itself.