10.6 Topology in the real line[2C6]

E296

[0S0] Show that a set \(A⊆ℝ\) is an interval if and only it is convex, if and only if it is connected.

(A part of the proof is in Theorem 5.11.3 in [ 2 ] ).

Hidden solution: [UNACCESSIBLE UUID ’0S1’]

(Note how in this case the exercises 1 and 8 coincide).

E296

[0S2] Let us fix \(𝛼∈ℝ\), consider the set \(A\) of numbers of the form \(𝛼 n + m\) with \(n,m\) integers. Show that \(A\) is dense in \(ℝ\) if and only if \(𝛼\) is irrational. Hidden solution: [UNACCESSIBLE UUID ’0S3’]

E296

[0S4]Given \(I⊆ ℚ\) non-empty, show that \(I\) is connected if and only \(I\) contains only one point. Hidden solution: [UNACCESSIBLE UUID ’0S5’]

E296

[0S6] Show that every open non-empty set \(A⊂ℝ\) is the union of a family (at most countable) of disjoint open intervals. Hidden solution: [UNACCESSIBLE UUID ’0S7’]

E296

[0S8] Find a compact \(A⊂ ℝ\) that has a countable number of accumulation points. Hidden solution: [UNACCESSIBLE UUID ’0S9’]

E296

[0SB]Prerequisites:2. Show that the set \(A⊂ ℝ\) defined by

\[ A=\{ 0\} ∪ \{ 1/n : n∈ℕ, n≥ 1\} ∪ \{ 1/n+1/m : n,m∈ℕ, n≥ 1, m≥ 1\} \]

is compact; identify its accumulation points.

Hidden solution: [UNACCESSIBLE UUID ’0SC’]

E296

[0SD] Difficulty:**. Let \(A⊂ℝ\). We recall that \(D(A)\) is the derivative of \(A\) (i.e. the set of accumulation points of \(A\)). Describe a closed set \(A\) such that the sets

\[ A, D(A),D(D(A)),D(D(D(A)))\ldots \]

are all different.

Hidden solution: [UNACCESSIBLE UUID ’0SF’]

E296

[0SG]Prerequisites:10, 13, 7, 286.Difficulty:**.

Find a subset A of \(ℝ\) such that the following 7 subsets of \(ℝ\) are all distinct:

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

Also prove that no other different sets can be created by continuing in the same way (also replacing \(ℝ\) with a generic metric space).

Hidden solution: [UNACCESSIBLE UUID ’0SH’]

E296

[0W6]Difficulty:**.Prove that it is not possible to write \(ℝ\), or an interval \(D ⊆ ℝ\), as a countable and infinite union of closed and bounded intervals, pairwise disjoint.

Hidden solution: [UNACCESSIBLE UUID ’0W7’] [UNACCESSIBLE UUID ’0W8’]

[UNACCESSIBLE UUID ’0SJ’][UNACCESSIBLE UUID ’0SK’]