ANA1_L14

Recall

Recall

• $f:X\to Y$, then $f$ is continuous $⟺$ $f^{-1}$(open set in $Y$) is open in $X$
• Definition of open sets and closed sets.

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Proof of the backward implication from the last claim (Rudin, 4.8) in L13

Proof of $⟸$
Given an arbitrary $ϵ$ challenge, we want $\delta >0$ such that $d(x,p)<\delta$ $⟹d(f(x),f(p))<ϵ$. This is the same as saying we want $\delta >0$ such that $x\in B_{\delta }(p)⟹f(x)\in B_{ϵ}(f(p))$.
We know that $f^{-1}(B_{ϵ}(f(p)))$ is open in $X$ and contains $p$. Importantly, it contains a ball with center $p$. Take $\delta$ to be the radius of this ball. ❏

Since $f^{-1}$ preserves complements, we also have $f^{-1}$(closed set in $Y$) is a closed set in $X$.

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Examples and observations on open and closed sets

Recall: $C$ is closed in $X$ $⟺$ $X\setminus C$ is open in $X$.

Example 1(Example).

For $X=\mathbb{R}$,

• $(0,1)$ is open, not closed
• $[0,1]$ is not open, is closed
• $(0,1]$ is neither
• $\mathbb{R},\varnothing$ are both open and closed

For $X=\mathbb{C}$,

• $(0,1)$ is not open, not closed
• $[0,1]$ is not open, is closed
• $\mathbb{R}$ is not open, is closed

In general,

• Any metric space is an open and closed subset of itself
• In any metric space $X$, a singleton set is closed in $X$.
• In any metric space $X$, a singleton set is open in $X$ iff its element is an isolated point of $X$.

Example 2(Example).

Let $X\equiv \{-1\}\bigcup \{x|x\ge 0\}$ be a metric space. Then, the subset $\{-1\}\subset X$ is both open and closed.

Example 3(Example).

It is quite easy to confuse closed sets with closed intervals in $\mathbb{R}$. Non-trivial examples of closed sets which are not closed intervals:

• The cantor set; it is an arbitrary intersection of closed sets, and hence is closed.
• Sets of the form $[x,\infty )$ and $(\infty ,x]$.

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An aside

The above examples motivate the notion of distance of a point from a set (you’ll see why).

Let $X$ be a metric space. Let $A\subset X$, and $p\in X$.
Define $d(p,A)\equiv \inf \{d(p,a)|a\in A\}$.

Now, $d(p,A)=0$ only when $p\in A$ or $p$ is a limit point of $A$.
Observe that
$A$ is closed $⟺$ ($d(p,A)=0⟺p\in A$ ). (here’s where you see why)

Now, fix $A$ and vary $p$. Then, is $d_A(p):X\to \mathbb{R}$ continuous? (Yes; prove it.)

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An equivalent property to being closed

Theorem 4(Theorem).

$C$ is closed in $X$ $⟺$ $C$ contains all of its limit points.

Proof of $⟹$
Let $q\in X\setminus C$. Since $X\setminus C$ is open, there exists $r>0$ such that $B_r(q)\subset X\setminus C$. So, $B_r(q)\cap C=\varnothing$, which implies $q$ is not a limit point of $C$. Therefore, any limit point must be in $C$. ❏

Proof of $⟸$
If $C$ contains all of its limit points, no point outside $C$ is a limit point of $C$. Want to show $X\setminus C$ is open. Take $p\in X\setminus C$. We know $p$ is not a limit point of $C$, so there should exist $r>0$ such that $B_r(p)\subset X\setminus C$. ❏

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The remnants of 2.18

Interior

Definition 5(Definition).

The interior of a set $S$, denoted $S$ is ${\bigcup }_{V\subset S}V$, where $V$ is an open set. It is the largest open set contained in $S$.

Closure

Definition 6(Definition).

The closure of $S$, denoted by $\bar{S}$, is ${\bigcap }_{C\supset S}C$, where $C$ are closed sets. It is the smallest closed set containing $S$.

Rudin defines (2.26) closure of $E$ as $E\cup E^{\prime }$, where $E^{\prime }$ is the set of all limit points of $E$. These definitions are, of course, equivalent.

Perfect sets

Defined perfect sets, and noted that they are not important for this course.

Definition 7(Definition).

$E$ is perfect if $E$ is closed and every point of $E$ is a limit point of $E$.

For example, the Cantor set is a perfect set.

Dense subsets

Definition 8(Definition).

$E$ is dense in $X$ if every point of $X$ is either a member of $E$ or a limit point of $E$ (or both).

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Open and closed sets under intersections and unions

Rudin, 2.24

Theorem 9(Theorem).

1. For any collection $\{U_{\alpha }\}$ of open sets, ${\bigcup }_{\alpha }{U}_{\alpha }$ is open.
2. For any collection $\{C_{\alpha }\}$ of closed sets, ${\bigcap }_{\alpha }{C}_{\alpha }$ is closed.
3. For any finite collection of open sets $\{U_1,U_2,\dots U_n\}$, ${\bigcap }_{i=1}^n{U}_i$ is open.
4. For any finite collection of closed sets $\{C_1,C_2,\dots C_n\}$, ${\bigcup }_{i=1}^n{C}_i$ is closed.

Proof of 1
An open set by definition is a union of open balls. It follows that a union of open sets is also a union of balls. ❏

Proof of 2
Follows from 1 and De Morgan’s laws.

Proof of 3
We will show that if $U_1$ and $U_2$ are open sets, $U_1\bigcap U_2$ is an open set. 3 follows by induction.
Let $p\in U_1\bigcap U_2$. There exist $r_1,r_2>0$ such that $B_{r_1}(p)\subset U_1$ and $B_{r_2}\subset U_2$. WLOG, let $r_1\le r_2$. It follows that $B_{r_1}(p)\subset B_{r_2}(p)$. Thus, $B_{r_1}\subset U_1$ and $B_{r_1}\subset U_2$, i.e, $B_{r_1}(p)\subset U_1\bigcap U_2$. ❏

Proof of 4
Follows from 3 and De Morgan’s laws.

Note that (3) is not true for the infinite case. Example: ${\bigcap }_{n=1}^{\infty }\left(-\frac{1}{n},\frac{1}{n}\right)=\{0\}$.
Note that (4) is not true for the infinite case. Example: ${\bigcup }_{n\ge 2}\left[\frac{1}{n},1-\frac{1}{n}\right]=(0,1)$.
^ Rudin 2.25

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Epilogue

Observed that
$p$ is an interior point of $S$ $⟺$ $S$ is a neighborhood of $p$.

Assigned reading: 2.18 - 2.29
Exercise: prove 2.30.