10.2 Topology in metric spaces[2C2]

Let \((X,d)\) be a metric space.

Note that, having the operational definition 282 of ”open set”, then the axioms (in the definition 243) in this case become theorems.

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[0NZ] Topics:balls.

Prove that

\begin{equation} \label{eq:inclpalle} B_{𝜌}(x)⊆ B_ r(x_ 0) \end{equation}

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for every \(x∈ B_𝜌(x_ 0)\) and for every \(0{\lt}𝜌≤ r-d(x,x_ 0)\). Hidden solution: [UNACCESSIBLE UUID ’0P0’]

E283

[0P1] Topics:balls, disks. Let \(x_ 1,x_ 2∈ X\), \(r_ 1,r_ 2{\gt}0\), if \(d(x_ 1,x_ 2)≥ r_ 1+r_ 2\) then

\begin{equation} \label{eq:disg_ palle} B_{r_ 1}(x_ 1)∩ D_{r_ 2}(x_ 2)=∅\quad . \end{equation}

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Hidden solution: [UNACCESSIBLE UUID ’0P2’]

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[0P3] Topics:interior. Prerequisites:1.Show that \(B_ r(x)\) is an open set using the definition 282. Hidden solution: [UNACCESSIBLE UUID ’0P4’]

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[0P5] Prove that a metric space is \(T_ 2\) i.e. Hausdorff (see definition in 245).

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[0P6] If \(A=B^ c\) then show that \((\overline B)^ c={{A}^\circ }\) (using the definitions in this section).

Hidden solution: [UNACCESSIBLE UUID ’0P7’]

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[0P8]Prerequisites:5.Show that the notions of interior and closure seen above are equivalent to those presented in the definition 243.

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[0PB] Topics:interior. Show that \(E\) is open if and only if \({{E}^\circ }=E\). Hidden solution: [UNACCESSIBLE UUID ’0PC’]

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[0PD] Topics:interior. Show that if \(A⊆ B⊆ X\) and \(A\) is open then \(A⊆ {{B}^\circ }\) using the above definitions.

Hidden solution: [UNACCESSIBLE UUID ’0PF’]

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[0PG] Topics:interior. Show that if \(A⊆ B⊆ X\) then \({{A}^\circ }⊆ {{B}^\circ }\). Hidden solution: [UNACCESSIBLE UUID ’0PH’]

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[0PJ] Topics:interior.Prerequisites:3,8.

Given \(X\) metric space and \(A⊆ X\), show that

\[ {{A}^\circ } = {{\left({{A}^\circ }\right)}^\circ }~ ~ , \]

using the above definitions.

For what has been said in 7, this is equivalent to saying that \({{A}^\circ }\) is an open set.

(For the case of \(X\) topological space, see the 5)

Hidden solution: [UNACCESSIBLE UUID ’0PK’]

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[0PM] Topics:interior.

Show that \(\overline E=E\) if and only if \(E\) is closed.

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[0PP] Topics:closure. Prerequisites:9,5.(Replaces 0PN)

Show that if \(B⊆ A⊆ X\) then \( \overline B⊆ \overline A\); using the above definitions, or by switching to complement set and using 9.

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[0PQ] Topics:closure.Prerequisites:5, 12.

Given a metric space \(X\) and a set \(A⊆ X\), show that

\[ \overline A= \overline{\left(\overline A\right)} \]

either by transitioning to the complement set and using 10, or by using the definition of \(\overline A\) as ”set of adherent points”.

As discussed in 11, this is equivalent to saying that \(\overline A\) is a closed set.

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[0PR] Let \(E⊆ X\), then \(E\) is a metric space with the restricted distance \(\tilde d=d|_{E× E}\).

Show that \(A⊆ E\) is open in \((E,\tilde d)\) (as defined at the beginning of this section) if and only there exists a set \(V⊆ X\) open in \((X,d)\) such that \(V∩ E = A\).

(The second way of defining ”open” is used in topology.)

Hidden solution: [UNACCESSIBLE UUID ’2GD’]

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[0PS] Prerequisites:2.Let \(X\) be a set with two distances \(d_ 1,d_ 2\); let’s call \(𝜏_ 1,𝜏_ 2\) respectively the induced topologies. We have that \(𝜏_ 1⊆ 𝜏_ 2\) if and only if

\[ ∀ x∈ X~ ∀ r_ 1{\gt}0~ ∃ r_ 2{\gt}0 ~ :~ B^ 2(x,r_ 2)⊆ B^ 1(x,r_ 1) \]

where

\[ B^ 2(x,r_ 2)=\{ y∈ X:d^ 2(x,y){\lt}r_ 2\} \quad ,\quad B^ 1(x,r_ 1)=\{ y∈ X:d^ 1(x,y){\lt}r_ 1\} \quad . \]

Note that this exercise is the analogue in metric spaces of the principle 2 for the bases of topologies.

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[0PT] Prerequisites:2, 15, 4, 2,3.

Having fixed \((X_ 1,d_ 1),\ldots ,(X_ n,d_ n)\) metric spaces, let \(X=X_ 1× \cdots × X_ n\).

Let \(𝜑\) be one of the norms defined in eqn. 12.1 in Sec. 12.1. Two possible examples are \(𝜑(x)=|x_ 1|+\cdots + |x_ n|\) or \(𝜑(x)=\max _{i=1\ldots n} |x_ i|\).

Finally, let’s define for \(x,y∈ X\)

\begin{equation} d(x,y)=𝜑\big( d_ 1(x_ 1,y_ 1), \ldots , d_ n(x_ n,y_ n)\big)\quad . \label{eq:dist_ prodotto} \end{equation}

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Show that \(d\) is a distance; show that the topology in \((X,d)\) coincides with the product topology (see 2).

Note that this approach generalizes the definition of the Euclidean distance between points in \(ℝ^ n\) (taking \(X_ i=ℝ\) and \(𝜑(z)=\sqrt{∑_ i |z_ i|^ 2}\)). We deduce that the topology of \(ℝ^ n\) is the product of the topologies of \(ℝ\).

Hidden solution: [UNACCESSIBLE UUID ’0PX’]

See also the exercise 2, which reformulates the above using the concept of bases of topologies.

[UNACCESSIBLE UUID ’0PV’]

[UNACCESSIBLE UUID ’0PW’] [0PY] Prerequisites:2,13.

Let \(D(x,r){\stackrel{.}{=}}\{ y\in X : d(x,y)\le r\} \) be the disk, show that it is closed.

Let \(S(x,r){\stackrel{.}{=}}\{ y\in X : d(x,y)= r\} \) be the sphere, show that it is closed.

Hidden solution: [UNACCESSIBLE UUID ’0PZ’] [0Q0]Prerequisites:286,3,286, 12.Let \(r{\gt}0\).

Let \(D(x,r){\stackrel{.}{=}}\{ y∈ X : d(x,y)≤ r\} \) be the disk; show that \(\overline{B(x,r)}⊆ D(x,r)\) and that \(B(x,r)⊆ {{D(x,r)}^\circ }\).

Let \(S(x,r){\stackrel{.}{=}}\{ y∈ X : d(x,y)= r\} \) be the sphere; show that \(∂{B(x,r)}⊆ S(x,r)\).

Find examples of metric spaces in which the above equalities (one, or both) do not hold.

Find an example of a metric space where there is a disk that is open ¹ .

(See also 1 for the case of space \(ℝ^ n\)). Hidden solution: [UNACCESSIBLE UUID ’0Q1’][UNACCESSIBLE UUID ’0Q2’] [0Q3] Prerequisites:6.Let \(A⊆ X\) where \((X,d)\) is a metric space, we have that \(x∈ ∂ A\) if and only if there exists \((y_ n)⊆ A\) and \((z_ n)⊆ A^ c\) sequences such that \(y_ n→ x\) and \(z_ n→ x\). Hidden solution: [UNACCESSIBLE UUID ’0Q4’] [0Q5] Prerequisites:Section  8.9. Find an example of a metric space \((M,d)\) that does not satisfy the second axiom of countability, i.e. such that there is no countable base for the topology associated with \((M,d)\).

Hidden solution: [UNACCESSIBLE UUID ’0Q6’] [0Q7] Prerequisites:Section  8.9.Let \((M,d)\) be a metric space and suppose that there exists \(D⊆ M\) that is countable and dense. Such \((M,d)\) is called separable. Show that \((M,d)\) satisfies the second axiom of countability.

The converse is true in any topological space, see 2. [0Q8] Prerequisites:8,12, 7, 5.Difficulty:*.

Let \(X\) be a metric space, and \(A⊆ X\). We want to study the ”open-close” operation \(\overline{({{ A}^\circ })}\) (which is the closure of the interior of \(A\)).

• Show a simple example where \(\overline{({{ A}^\circ })}\) is not contained \(A\).

• Then write a characterization of \(\overline{({{ A}^\circ })}\) using sequences and balls.

• Use it to show that the ”open-close” operation is idempotent, that is, if \(D=\overline{({{ A}^\circ })}\) and then \(E=\overline{({{D}^\circ })}\) then \(E=D\).

Hidden solution: [UNACCESSIBLE UUID ’0Q9’][UNACCESSIBLE UUID ’0QB’] [0QC] Prerequisites:1.Show that, for every closed set \(C⊆ X\) there exist countably many open sets \(A_ n\) such that \(⋂_ n A_ n=C\).

Hidden solution: [UNACCESSIBLE UUID ’0QD’]

A set obtained as an intersection of countably many open sets is known as ”a \(G_𝛿\) set”. The previous exercise shows that in a metric space every closed is a \(G_𝛿\).

Passing to the complement set, one obtains this statement. A set that is union of countably many closed sets is known as ”an \(F_𝜎\) set”. The previous exercise shows that in a metric space every open set is an \(F_𝜎\) set.

See also the section 14.4. [0QF] Difficulty:**.Find an example of a metric space where for every \(x∈ X,r{\gt}0\), \(B_ r(x)\) is a closed set, but the associated topology is not discrete.  ²

Hidden solution: [UNACCESSIBLE UUID ’0QG’]

We note that such a space must be totally disconnected as shown in 1.

[UNACCESSIBLE UUID ’0QH’]

Bases composed of balls

To face these exercises it is necessary to know the concepts seen in Sec. 8.8.

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[0QJ]Prerequisites:2, 2.Show that the intersection of two balls is an open set (according to the definition 281). Hence the family of all balls meets the requirements (a) and (b) in exercise 2; so (as shown in 2), the family of balls is a base for the topology that it generates (which is the topology associated with metric space).

Hidden solution: [UNACCESSIBLE UUID ’0QK’]

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[0QM]

Let’s review the exercise 16.

Having fixed \((X_ 1,d_ 1),\ldots ,(X_ n,d_ n)\) metric spaces, let \(X=X_ 1× X_ 1× \cdots × X_ n\).

Let \(d\) be the distance

\[ d(x,y)=\max _{i=1,\ldots n} d_ i(x_ i,y_ i)~ ~ . \]

This is the same \(d\) defined as in eqn. ?? inside 16, setting \(𝜑(x)=\max _{i=1\ldots n}|x_ i|\). We indicate with \(B^ d(x,r)\) the ball in \((X,d)\) of center \(x∈ X\) and radius \(r{\gt}0\).

We want to show that \(d\) induces the product topology on \(X\), using the results seen in Sec. 8.8.

Taken \(t∈ X_ i, r{\gt}0\) we indicate with \(B^{d_ i}(t,r)\) the ball in metric space \((X_ i,d_ i)\). Let \({\mathcal B}_ i\) be the family of all balls in \((X_ i,d_ i)\).

Let \(\mathcal B\) be defined as

\[ {\mathcal B}=\left\{ ∏_{i=1}^ n B^{d_ i}(x_ i,r_ i) : ∀ i,x_ i∈ X_ i,r_ i{\gt}0\right\} \]

This is the same \({\mathcal B}\) defined in 2.

Show that every ball \(B^ d(x,r)\) in \((X,d)\) is the Cartesian product of balls \(B^{d_ i}(x_ i,r)\) in \((X_ i,d_ i)\). So let \(\mathcal P\) be the family of balls \(B^ d(x,r)\) in \((X,d)\).

From 1 we know that \(\mathcal P\) is a base for the standard topology in the metric space \((X,d)\).

Use 2 to show that \(\mathcal P\) and \({\mathcal B}\) generate the same topology \(𝜏\).

Use 2 to prove that \(𝜏\) is the product topology.

We conclude that the distance \(d\) generates the product topology.

Accumulation points, limit points

Let’s redefine this notion (a special case of what we saw in 250)

The set of accumulation points of \(A\) is called derived set, we will indicate it with \(D(A)\).

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[0QP] Topics:adherent point, accumulation point.

Check that

• Each accumulation point is also an adherent point, in symbols \(D(A)⊆ \overline A\);

• if a point adhering to \(A\) is not in \(A\) then it is an accumulation point;

So we obtain that \(\overline A=A∪ D(A)\). [UNACCESSIBLE UUID ’0QQ’]

[0QR] Given \(A⊆ X\), a point \(x∈ X\) is an accumulation point if and only if there exists a sequence \((x_ n)⊆ A\) which is injective and such that \(\lim _{n→ ∞} x_ n=x\). [0QS]Let \((X,d)\) metric space, and \(x∈ X\). Show that \(A=\{ x\} \) is closed; and that \(A\) has an empty inner part if and only if \(x\) is accumulation point. Hidden solution: [UNACCESSIBLE UUID ’0QT’] [0QV]Let \(A⊆ X\) and let \(D(A)\) be the derivative (i.e. the set of its accumulation points). Show that \(D(A)\) is closed. Hidden solution: [UNACCESSIBLE UUID ’0QW’]

Let’s add this definition (a special case of 262).

In English literature the terms ”cluster point”, ”limit point” and ”accumulation point” are sometimes considered synonimous, which can be confusing. We will stick to the proposed definitions 287 and 288.

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[0QY] Find an example of a metric space \((X,d)\) and a bounded sequence \((x_ k)_ k⊆ X\) that has a single limit point \(x\) but that does not converge.

See also 2.

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[0QZ] Prerequisites:2,9.

• If a sequence \((a_ k)_ k⊆ X\) converges to \(x\) then it has an unique limit point, which is \(x\).

• If a Cauchy sequence \((a_ k)_ k⊆ X\) has a limit point then there is only one limit point \(x\) and \(\lim _ k a_ k=x\).

Hidden solution: [UNACCESSIBLE UUID ’0R0’]

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[2F3]Topics:perfect set.Prerequisites:1,117,266.Difficulty:**.

Suppose \((X,d)\) is a complete metric space. A closed set \(E⊆ X\) without isolated points, i.e. consisting only of accumulation points, is called a perfect set.

Let \(C\) be the Cantor set. Assume that \(E\) is perfect and non-empty. Show that there exists a continuous function \(𝜑:C→ E\) that is an homeomorphism with its image. This implies that \(|E|≥ |ℝ|\).

So, in a sense, any non-empty perfect set contains a copy of the Cantor set.

This can be proven without relying on continuum hypothesis 117. Cf. 4.

Due to 4, it is enough to show that there exists a \(𝜑:C→ E\) continuous and injective.

Hidden solution: [UNACCESSIBLE UUID ’2F4’]

[UNACCESSIBLE UUID ’0R1’]

Other exercises on these topics are 5, 6, 7, 2 and 2.