: Bases<a class="btn btn-sm" href="/UUID/EDB/2B5/?lang=eng"><span class="ttfamily"><small class="scriptsize">[2B5]</small></span></a>

8.8 Bases[2B5]

Definition 265 Base

[0KK] Given a topological space \((X,𝜏)\), a base ¹ is a collection \({\mathcal B}\) of open sets (i.e. \({\mathcal B}⊆ 𝜏\)) with the property that every element of \(𝜏\) is an union of elements of \({\mathcal B}\).

For example, if \(X\) is a metric space, then the family of all open balls is a base.

E265

[0KM] Let \({\mathcal B}\) be a base for a topology \(𝜏\) on \(X\); chosen an open set \(A∈ 𝜏\), for every \(x∈ A\) we can choose a \(B_ x∈ {\mathcal B}\) with \(x∈ B_ x\), and such that \(A=⋃_{x∈ A}B_ x \).

Hidden solution: [UNACCESSIBLE UUID ’0KN’][UNACCESSIBLE UUID ’0KP’]

E265

[0KQ] Prerequisites:1.Let \({\mathcal B}\) be a base for a topology \(𝜏\) on \(X\). Show that, given \(x∈ X\),

\[ \{ B∈ {\mathcal B} : x∈ B\} \]

is a fundamental system of neighbourhoods for \(x\) . [UNACCESSIBLE UUID ’0KR’]

[0KS] Prerequisites:2, 2, 1.Let \({\mathcal B}\) be a base for a topology \(𝜏\) on \(X\). Show that, for any given \(A⊆ X\),

\[ {{A}^\circ } = {\underline⋃} \{ B∈{\mathcal B}: B⊆ A \} \]

while

\[ \overline A = \{ x∈ X: ∀ B∈{\mathcal B} , x∈ B⇒ B∩ A≠∅ \} \]

Hidden solution: [UNACCESSIBLE UUID ’0KT’] [0M7] Prerequisites:1.Given \(X\), given a base \({\mathcal C}\) for a topology \(𝜎\) on \(X\), and a base \({\mathcal B}\) for a topology \(𝛽\) on \(X\), we have that \(𝜎⊇𝛽\) if and only if for every \(x∈ X\) and for every \(B∈ {\mathcal B},B∋ x\) there exists \(C∈ {\mathcal C},C∋ x, C⊆ B\). Hidden solution: [UNACCESSIBLE UUID ’0M8’] [0KV]Prerequisites:1.Let \(X=\{ 1,2,3\} \) and let \({\mathcal B}=\{ \{ 1,2\} ,\{ 2,3\} \} \); let \(𝜏\) be the smallest topology that contains \({\mathcal B}\), show that \({\mathcal B}\) is not a base for \(𝜏\).

Hidden solution: [UNACCESSIBLE UUID ’0KW’]

It is therefore interesting to try to understand when a family \({\mathcal B}\) can be the base for a topology. [0KX]Let \({\mathcal B}\) be a base for a topology \(𝜏\) on \(X\); then the following two properties apply.

(a): \(\underline⋃ {\mathcal B} = X\) that is, the union of all the elements of the base is \(X\).
(b): Given \(B_ 1,B_ 2∈ {\mathcal B}\) for each \(x∈ B_ 1∩ B_ 2\) there exists \(B_ 3∈ {\mathcal B}\) such that \(x∈ B_ 3⊆ B_ 1∩ B_ 2\).

Hidden solution: [UNACCESSIBLE UUID ’0KY’] [0KZ] Prerequisites:1,4.Conversely, let \(X\) be a set and \({\mathcal B}\) a family of subsets that verify the previous properties (a),(b) seen in 2. Let \(𝜎\) the family of sets that are obtained as a union of elements of \({\mathcal B}\), in symbols ²

\[ 𝜎{\stackrel{.}{=}}\left\{ ⋃_{i∈ I} A_ i : I~ \text{ family of indexes and } ~ A_ i∈ {\mathcal B} ∀ i∈ I\right\} ~ ~ ; \]

it is meant that also \(∅∈ 𝜎\). Show that \(𝜎\) is a topology.

Hidden solution: [UNACCESSIBLE UUID ’0M0’] [0M1] Prerequisites:Generated topology 1, 2, 2.Let’s resume 2. Let again \(X\) be a set and \({\mathcal B}\) a family of subsets that satisfy the above properties (a),(b) seen in 2; suppose \(𝜏\) the smallest topology that contains \({\mathcal B}\). Prove that \({\mathcal B}\) is a base for \(𝜏\).

Hidden solution: [UNACCESSIBLE UUID ’0M2’]

We can therefore say that a family that satisfies (a),(b) is a base for the topology it generates. This answers the question posed in 2. [0M3]Prerequisites:1,2,2. Let now \(X_ 1,\ldots X_ n\) be topological spaces with topologies, respectively, \(𝜏_ 1,\ldots 𝜏_ n\); let \(X=∏_{i=1}^ nX_ i\) be the Cartesian product. We apply the above results to define the product topology \(𝜏\): this can be described in two equivalent ways.

Union of all Cartesian products of open sets ³

\begin{align*} 𝜏=\Big\{ ⋃_{j∈ J} ∏_{i=1}^ n A_{i,j} : A_{1,j}∈𝜏_ 1,\ldots A_{n,j}∈𝜏_ n∀ j∈ J , J~ \\ \text{arbitrarily chosen sets of indexes} \Big\} ~ ~ . \end{align*}
\(𝜏\) is the smallest topology that contains Cartesian products of open sets.

Hidden solution: [UNACCESSIBLE UUID ’0M4’] [0M5]Prerequisites:2,2,2.Let now \(X_ 1,\ldots X_ n\) be topological spaces with topologies \(𝜏_ 1,\ldots 𝜏_ n\) respectively and suppose that \({\mathcal B}_ 1,{\mathcal B}_ 2,\ldots {\mathcal B}_ n\) are bases for these spaces. Let \(X=∏_{i=1}^ nX_ i\) be the Cartesian product, and let

\[ {\mathcal B}=\left\{ ∏_{i=1}^ n A_ i : A_ 1∈{\mathcal B}_ 1,A_ 2∈{\mathcal B}_ 2,\ldots A_ n∈{\mathcal B}_ n\right\} \]

The family of all cartesian products of elements chosen from their respective bases. Show that \({\mathcal B}\) is a base for the product topology. (This exercise generalizes the previous 2, taking \({\mathcal B}_ i=𝜏_ i\)).

Hidden solution: [UNACCESSIBLE UUID ’0M6’]

See also the exercise 2 for an application to the case of metric spaces. [2F7]Prerequisites:2,2,2.

Let, more in general, \(I\) be a non-empty index set, and let \((X_ i,\tau _ i)\) be topological spaces, for \(i\in I\); let \({\mathcal B}_ i\) be a base for \(\tau _ i\). (Note that the choice \({\mathcal B}_ i=\tau _ i\) is allowed.)

Let \(X=∏_{i\in I} X_ i\) be the Cartesian product.

We define the product topology \(𝜏\) on \(X\), similarly to 2, but with a twist.

A base \({\mathcal B}\) for \(\tau \) is the family of all sets of the form \(A=∏_{i\in I} A_ i\) where

\[ \forall i\in I, A_ i\in {\mathcal B}_ i\lor A_ i= X_ i~ ~ , \]

and moreover \(A_ i= X_ i\) but for finitely many \(i\).

Show that \({\mathcal B}\) satisfies the requirements in 2, so it is a base for the topology \(\tau \) that it generates. Show that the product topology does not depend on the choice of the bases \({\mathcal B}_ i\).

Hidden solution: [UNACCESSIBLE UUID ’2F8’] [0M9] Prerequisites:55, 2. We verify that what is expressed in 8 also applies to the ”base”. Let \({{\mathcal B}}\) be a base for a topology \(𝜏\) on \(X\); consider the descending order between sets (formally \(A ⪯ B \iff A⊇ B\)); with this order \(({{\mathcal B}},⪯)\) is a directed set, whose minimum is \(∅\). Now suppose the topology is Hausdorff. Then taken \(x∈ X\), let \({{\mathcal U}}=\{ A∈{{\mathcal B}}: x∈ A\} \) be the family of elements of the base that contain \(x\): show that \({{\mathcal U}}\) is a directed set. Show that it has minimum if and only if the singleton \(\{ x\} \) is open. Hidden solution: [UNACCESSIBLE UUID ’0MB’] [2F5] Consider a totally ordered set \(X\) (that has at least two elements), and the family \(\mathcal F\) of all open-ended intervals

\begin{align} (x,∞) {\stackrel{.}{=}}\{ z∈ X : x{\lt}z\} ~ ~ ,~ ~ (-∞,y){\stackrel{.}{=}}\{ z∈ X : z{\lt}y\} ~ ~ ,\nonumber \\ ~ ~ (x,y){\stackrel{.}{=}}\{ z∈ X : x{\lt}z{\lt}y\} \label{eq:intervalli_ topologia_ ordine} \end{align}

for all \(x,y∈ X\). (Cf. 68.) Prove that this is a base for a topology, i.e. that it satisfies 2. So \(\mathcal F\) is a base for the topology \(𝜏\) that it generates. This topology \(𝜏\) is called order topology.

If \(X\) has no maximum and no minimum, then only the intervals \((x,y)\) are needed to form a base for \(𝜏\). This is the case for the standard topologies on \(ℝ\), \(ℚ\), \(ℤ\), [2FD]Prerequisites:2.Consider topological spaces \((X_ i,\tau _ i)\), each with the discrete topology (and each \(X_ i\) has at least two elements). Let \(I=ℕ\) or \(I=\{ 0,1,\ldots N\} \); let \(X=∏_{i∈ I} X_ i\) be the Cartesian product. We define the product topology \(𝜏\) on \(X\), as explained in 2. Describe a simple base for this topology. Moreover, if \(I=ℕ\), show that the topology \(\tau \) is not the discrete topology.

Hidden solution: [UNACCESSIBLE UUID ’2FF’]

[2F9]Prerequisites:2,2,64,2.

Consider totally ordered sets \((X_ i,≤_ i)\) (each has at least two elements), and the associated order topologies \(𝜏_ i\).

Let \(I=ℕ\) or \(I=\{ 0,1,\ldots N\} \); let \(X=∏_{i∈ I} X_ i\) be the Cartesian product.

Consider these two topologies.

We define the product topology \(𝜏\) on \(X\), as explained in 2.
We order \(X\) with the lexicographical order \(⪯\), and then we build the order topology \(𝜎\) on \(X\). (See 64,2)

Is there an inclusion between \(𝜎\) and \(𝜏\)?

If every \(X_ i\) is finite, prove that these two topologies coincide ⁴ .

Hidden solution: [UNACCESSIBLE UUID ’2FC’]