10.5 Connected set[2C5]
See definitions in Sec. 8.5. We also define this notion.
[0RG]A topological space \((X,𝜏)\) is ”path connected” if, for every \(x,y∈ X\), there is a continuous arc \(𝛾:[a,b]→ X\) with \(x=𝛾(a),y=𝛾(b)\).
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[0RH] Find a sequence of connected closed sets \(C_ n⊆ℝ^ 2\) such that \(C_{n+1}⊆ C_ n\) and the intersection \(⋂_ nC_ n\) is a non-empty and disconnected set.
Can you find such an example in \(ℝ\)?
Hidden solution: [UNACCESSIBLE UUID ’0RJ’]
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[0RK] Find a sequence of sets \(C_ n⊆ℝ^ 2\) that are closed and path connected, such that \(C_{n+1}⊆ C_ n\) and the intersection \(⋂_ nC_ n\) is non-empty, connected, but not path connected.
Hidden solution: [UNACCESSIBLE UUID ’0RM’][UNACCESSIBLE UUID ’0RN’]
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[0RP] Consider the example of the set \(E⊆ℝ^ 2\) given by
\begin{equation} E=\big\{ (0,t):\ -1≤ t≤1\big\} ∪\Big\{ \Big(x,\sin \frac 1 x\Big):\ x∈ (0,1]\Big\} \quad .\label{eq:connesso_ ma_ non_ archi} \end{equation}296Show that this set is closed, connected, but is not path connected.
Hidden solution: [UNACCESSIBLE UUID ’0RQ’]
This set is sometimes called closed topologist’s sine curve [ 45 ] .
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[0RR] Difficulty:*.Let \((X,d)\) be a metric space. Show that \(E⊆ X\) is disconnected if and only if ”there are two disjoint open sets, each of which intersect \(E\) and such that \(E\) is covered by their union” (see the proposition formalized in eqn. ?? in the exercise 5).
Hidden solution: [UNACCESSIBLE UUID ’0RS’]
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[0RT]Let \(D⊆ ℝ^ 2\) be countable; show that \(ℝ^ 2⧵ D\) is path connected.
Hidden solution: [UNACCESSIBLE UUID ’0RV’]
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[0RY]Find an example of a metric space \(X\) that is path connected, where there exists an open subset \(A⊆ X\) that is connected but not path connected. Hidden solution: [UNACCESSIBLE UUID ’0RZ’]