8 Topology[0G5]
Let \(X\) be a fixed and non-empty set. We will use this notation. For each set \(A⊆ X\) we define that \(A^ c=X⧵ A\) is the complement to A.
[2DY] A topological space is a pair \((X,𝜏)\) where \(X\) is a non-empty set with associated the family \(𝜏\) of the open sets, which is called topology.
[0G6] A topology \(𝜏⊆{\mathcal P}(X)\) is a family of subsets of \(X\) that are called open sets. This family enjoys three properties: \(∅,X\) are open; the intersection of a finite number of open sets is an open sets; the union of an arbitrary number of open sets is an open set.
[0G7] Let \(A,B⊆ X\) be two subsets.
The interior of \(A\), denoted by \({{A}^\circ }\), is the union of all the open sets contained in \(A\), and therefore is the biggest open set contained in \(A\);
the closure of \(B\), denoted by \(\overline{B}\), is the intersection of all the closed sets that contain \(B\), i.e. is the smallest closed that contains \(B\).
We say that \(A\) is dense in \(B\) if \(\overline A ⊇ B\). 1
The boundary \(∂ A\) of \(A\) is \(∂ A=\overline A⧵ {{A}^\circ }\).
[0G8] A topological space \((X,𝜏)\) is said to be \(T_ 2\), or ”Hausdorff space”, if \(∀ x,y ∈ X\) exist \(U,V∈ 𝜏\) open disjoint with \(x∈ U,y∈ V\).
[2F6]Any set \(X\) can be endowed with many different topologies. Here are two simple examples:
When a set \(X\) is endowed with the discrete topology, then all sets are open, and therefore closed. Equivalently, the discrete topology is caracterized by: every singleton is an open set.
When a set \(X\) is endowed with the indiscrete topology, then the only open (and, closed) sets are \(X,\emptyset \).
Further informations on these subjects may be found in Chap. 2 of [ 22 ] or in [ 14 ] .
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[0G9]Show that if the space is \(T_ 2\) then every singleton \(\{ x\} \) is closed.
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[0GB]Show that if \(A⊆ B\) then \(\overline A⊆ \overline B\) and \({{A}^\circ }⊆ {{B}^\circ }\)
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[0GC]Show that if \(A=B^ c\) then \((\overline B)^ c={{A}^\circ }\), using the definitions 243 and 244.
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[0GD]Note that \(A⊇ {{A}^\circ }\) and \(B⊆ \overline B\), generally. Show that \(A\) is open if and only if \(A={{A}^\circ }\); and that \(B\) is closed if and only if \(B=\overline B\), using definitions 243 and 244.
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[0GF] Topics:interior. Given \(X\), a topological space, and \(A⊆ X\), show that
\[ {{A}^\circ } = {{\left({{A}^\circ }\right)}^\circ }~ ~ . \]using the definition of \({{A}^\circ }\) given above.
(For the case of \(X\) metric space, see also 10)
Hidden solution: [UNACCESSIBLE UUID ’0GG’]
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[0GH] Topics:closing. Given \(X\) topological space and \(A⊆ X\), show that
\[ \overline A= \overline{\left(\overline A\right)} \]or by switching to complement with respect to 5, and using the definition of \(\overline A\) like ”intersection of all the locks they contain \(A\)”.
(For the case of \(X\) metric space, see also 13)
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[0GJ] Topics:closure, interior. Let \(X\) be a topological space and \(A⊆ X\) open.
Show that \(A⊆ {{\left(\overline A\right)}^\circ }\) (the interior of the closure of \(A\)).
Find an example of an open set \(A⊂ℝ\) for which \(A≠ {{\left(\overline A\right)}^\circ }\).
Then formulate a similar statement for \(A\) closed, transitioning on to the complement.
Hidden solution: [UNACCESSIBLE UUID ’0GK’]
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[0GM]Given the sets \(A, \, B\subseteq {\mathbb {R}}\), determine the relations between the following pairs of sets
\begin{align*} \overline{A\cup B} \qquad \qquad \text{ \text{and}~ }& \, \qquad \qquad \overline{A}\cup \overline{B}\, ,\\ \overline{A\cap B} \qquad \qquad \text{ \text{and}~ }& \, \qquad \qquad \overline{A}\cap \overline{B}\, ,\\ {{({A\cup B})}^\circ } \qquad \qquad \text{ \text{and}~ }& \, \qquad \qquad {{A}^\circ } \cup {{B}^\circ }\, , \\ {{({A\cap B})}^\circ } \qquad \qquad \text{ \text{and}~ }& \, \qquad \qquad {{A}^\circ } \cap {{B}^\circ }\, . \end{align*}
Let \((X,\tau )\) be a topological space. Consider the descending ordering between sets 2 , with this ordering \(𝜏\) is a directed set; we note that it has minimum, given by \(∅\).
Now suppose the topology is Hausdorff. Then taken \(x∈ A\), let \({{\mathcal U}}=\{ A∈𝜏: x∈ A\} \) be the family of the open sets 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 (and in this case the minimum is \(\{ x\} \)). 3
Hidden solution: [UNACCESSIBLE UUID ’0GR’]
By the exercise 3, when \(\{ x\} \) is not open then \({{\mathcal U}}\) is a filtering set, and therefore can be used as a family of indices to define a nontrivial ”limit” (see Remark 240). We will see applications in section 8.7.
[0GS]Note:Written exam of 25 March 2017.Let \((X, τ )\), \((Y , θ) \) be two topological spaces with non-empty intersection and assume that the topologies restricted to \(C=X ∩ Y\) coincide (i.e. \(τ_{| C} = θ_{| C} \)) 4 and that \(C\) is open in both topologies (i.e. \(C∈ τ, C∈ θ\)). Prove that there is only one topology \(σ\) on \(Z=X ∪ Y\) such that \(σ_{| X} = τ\) and \(σ_{|Y} = θ\) and that \(X,Y∈ σ\). Hidden solution: [UNACCESSIBLE UUID ’0GT’][UNACCESSIBLE UUID ’0GV’]