Closure of Real Interval

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Theorem

Let $I$ be a nonempty real interval such that:

  • $I = \left({a \,.\,.\, b}\right)$
  • $I = \left[{a \,.\,.\, b}\right)$
  • $I = \left({a \,.\,.\, b}\right]$ or
  • $I = \left[{a \,.\,.\, b}\right]$.


Then $\operatorname{cl} \left({I}\right)$, the closure of $I$, is the closed interval $\left[{a \,.\,.\, b}\right]$.


Proof

Let $I$ be one of the intervals as specified in the exposition.


Note that:

$(1): \quad$ By Condition for Point being in Closure, $x \in \operatorname{cl} \left({I}\right)$ iff every open set in $\R$ containing $x$ contains a point in $I$.
$(2): \quad$ From Union of Open Sets of Metric Space, every open set in $\R$ is a union of open intervals.
Thus we also have that $x \in \operatorname{cl} \left({I}\right)$ iff every open intervals containing $x$ also contains a point in $I$.


This equivalence will be made use of througout.


Lemma: $x \in \left[{a \,.\,.\, b}\right] \implies x \in \operatorname{cl} \left({I}\right)$

Let $x \in \left[{a \,.\,.\, b}\right]$.

Let $\left({c \,.\,.\, d}\right)$ be an open interval in $\R$ such that $x \in \left({c \,.\,.\, d}\right)$.

We must show that $\left({c \,.\,.\, d}\right)$ contains a point in $I$.


One of the following three possibilities holds:

  • $a < x < b$
  • $x = a$
  • $x = b$


Case: $a < x < b$

In this case, $x \in I$ and $x \in \left({c \,.\,.\, d}\right)$, so $\left({c \,.\,.\, d}\right)$ contains a point in $I$.


Case: $x = a$

If $I$ contains $a$, then we are done since this means $x \in I$. So, assume that $a \notin I$.

Since $I$ is nonempty but does not contain $a$, we must have $a < b$.

Let $r$ be the minimum of $d$ and $b$, so that $r \le d$ and $r \le b$.

Since $a = x < d$ by choice of $d$ and since $a < b$ by assumption, we must have $a < r$.

Thus, by Reals are Close Packed, there exists some $s \in \R$ such that $a < s < r$.

To summarize, we have $c < x = a < s < r$, where $r \le d$ and $r \le b$.

This means that $s$ satisfies both $c < s < d$ and $a < s < b$.

Hence, $s$ is a point in $\left({c \,.\,.\, d}\right)$ which is also in $I$. The existence of such a point is what we wanted to show.


Case: $x = b$

This case is analogous to case when $x = a$.

Here we instead let $l$ be the maximum of $c$ and $a$, and select an $s$ such that $l < s < x = b < d$, where $c \le l$ and $a \le l$.


$\Box$


Lemma: $x \notin \left[{a \,.\,.\, b}\right] \implies x \notin \operatorname{cl} \left({I}\right)$

Suppose $x \notin \left[{a \,.\,.\, b}\right]$. We must find an open interval containing $x$ which does not contain a point in $I$.

There are two possibilities:

  • $x < a$ or
  • $x > b$


Case: $x < a$

By Reals are Close Packed, there exists $r \in \R$ such that $x < r < a$.

Thus $\left({x-1 \,.\,.\, r}\right)$ is an open interval, all of whose elements are less than $a$, and hence not in $I$.


Case: $x > b$

This is similarly to the case when $x < a$.

Here instead we pick $r$ such that $b < r < x$, and consider the interval $\left({r \,.\,.\, x+1}\right)$.


$\Box$


By the two lemmas proven above, $\left[{a \,.\,.\, b}\right] = \operatorname{cl} \left({I}\right)$.


$\blacksquare$


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