10.7 Topology in Euclidean spaces [2C7]
In the following we consider the metric space \({\mathbb {R}}^ n\) with the usual Euclidean distance.
- E296
[0SM] Prerequisites:286.Let \(B(x,r){\stackrel{.}{=}}\{ y∈ ℝ^ n : |x-y|{\lt} r\} \) be the ball; let \(D(x,r){\stackrel{.}{=}}\{ y∈ ℝ^ n : |x-y|≤ r\} \) the disc; let \(S(x,r){\stackrel{.}{=}}\{ y∈ ℝ^ n : |x-y|= r\} \) be the sphere. Show that \(\overline{B(x,r)}= D(x,r)\), that \(B(x,r)= {{D(x,r)}^\circ }\), and that \(∂{B(x,r)}= S(x,r)\). Also show that \(B(x,r)\) is not closed and \(D(x,r)\) is not open.
(This result holds more generally in any normed space: see 4).
- E296
[0SN] Prerequisites:2, 6. Given a sequence \((x_ k)_ k⊆ ℝ^ n\), these facts are equivalent
- a
the sequence is bounded and has a single limit point \(x\)
- b
\(\lim _ k x_ k=x\).
Hidden solution: [UNACCESSIBLE UUID ’0SP’] See also 1.
- E296
[0SQ] Prerequisites:6, 7.For each \(A⊆ ℝ^ n\) closed non-empty set, there exists \(B⊆ A\) such that \(A=∂ B\).
In which cases does there exist such a \(B\) that is countable?
In which cases does there exist such a \(B\) that is closed?
Hidden solution: [UNACCESSIBLE UUID ’0SR’][UNACCESSIBLE UUID ’0SS’]
See also 2.
Can you find it \(F⊆ E\)?
[UNACCESSIBLE UUID ’0SW’] Hidden solution: [UNACCESSIBLE UUID ’0SX’][UNACCESSIBLE UUID ’0SY’] [UNACCESSIBLE UUID ’0SZ’] [0T0]What are the sets \(A\subset {\mathbb {R}}^ n\) that are both open and closed?
Hidden solution: [UNACCESSIBLE UUID ’0T2’] [0T3] Let \(f:ℝ →ℝ^ n\) continue; show that these two conditions are equivalent
\(\lim _{t→∞} |f(t)|=+∞\) and \(\lim _{t→-∞} |f(t)|=+∞\);
\(f\) is proper, i.e. for every compact \(K⊂ℝ^ n\) we have that the counterimage \(f^{-1}(K)\) is a compact of \(ℝ\).
[0T4] Prerequisites:Section 8.9.Show that \(ℝ^ N\) satisfies the second axiom of countability. [0T5] Prerequisites:1. Note:exercise 4 in the written exam of 13/1/2011.
If \(A⊆ ℝ^ n\) is composed only of isolated points, then \(A\) has countable cardinality.
Conversely, therefore, if \(A⊆ ℝ^ n\) is uncountable then the derivative \(D(A)\) is not empty.
Hidden solution: [UNACCESSIBLE UUID ’0T6’] [0T7] Let \(A⊂ℝ^ n\) be a bounded set. For every \(\varepsilon {\gt}0\) there is a set \(I⊂ A\) that satisfies:
\(I\) is a finite set,
\(∀ x,y∈ I\), \(x≠ y\) you have \(x∉ B(y,\varepsilon )\) (i.e. \(d(x,y)≥ \varepsilon \)),
- \[ A⊆ ⋃_{x∈ I} B(x,\varepsilon )~ ~ . \]
Hidden solution: [UNACCESSIBLE UUID ’0T8’] [0T9]Difficulty:*.What is the cardinality of the family of open sets in \(ℝ^ n\)?
Hidden solution: [UNACCESSIBLE UUID ’0TB’]
[UNACCESSIBLE UUID ’0TC’] [0TD] Let \(E⊆ ℝ^ n\) be not empty and such that every continuous function \(f:E→ℝ\) admits maximum: show that \(E\) is compact.
(See 9 for generalization to metric spaces)
Hidden solution: [UNACCESSIBLE UUID ’0TF’]