数学 - Python - 解析学 - 距離空間の世界 - 位相の基礎的諸概念 - 閉包の内部の閉包の内部、内部の平方の内部の閉包、等式の証明

解析入門(中)
(松坂和夫 数学入門シリーズ 5)
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  • Pythonからはじめる数学入門 (楽天ブックス Yahoo!)

解析入門(中) (松坂和夫 数学入門シリーズ 5) (松坂 和夫(著)、岩波書店)の第12章(距離空間の世界)、12.1(位相の基礎的諸概念)、問題10の解答を求めてみる。

10. 
    
    $\begin{array}{l}{A}^{aiai}\\ =A^{aiaa}\backslash A^{aiaf}\\ =A^{\alpha ia}\backslash A^{aiaf}\\ =A^{aii}\\ =A^{ai}\end{array}$

    $\begin{array}{l}{A}^{iai}\subset A^{ia}\\ A^{iaia}\subset A^{iaa}\\ A^{iaia}\subset A^{ia}\\ A^{ia}\supset A^i\\ A^{iai}\supset A^{ii}=A^i\\ A^{iaia}\supset A^{ia}\\ A^{iaia}=A^{ia}\end{array}$

    （証明終）

コード

#!/usr/bin/env python3
from unittest import TestCase, main
import random
from sympy import Interval

print('10.')


class Test(TestCase):
    def test_boundary_intersect_closure_union(self):
        for _ in range(10):
            a0, a1 = [random.randrange(-10, 11) for _ in range(2)]
            A = Interval(a0, a1)
            self.assertEqual(
                A.closure.interior.closure.interior,
                A.closure.interior,
            )
            self.assertEqual(
                A.interior.closure.interior.closure,
                A.interior.closure,
            )
            A = Interval(a0, a1)
            self.assertEqual(
                A.closure.interior.closure.interior,
                A.closure.interior,
            )
            self.assertEqual(
                A.interior.closure.interior.closure,
                A.interior.closure,
            )
            A = Interval(a0, a1)
            self.assertEqual(
                A.closure.interior.closure.interior,
                A.closure.interior,
            )
            self.assertEqual(
                A.interior.closure.interior.closure,
                A.interior.closure,
            )
            A = Interval(a0, a1)
            self.assertEqual(
                A.closure.interior.closure.interior,
                A.closure.interior,
            )
            self.assertEqual(
                A.interior.closure.interior.closure,
                A.interior.closure,
            )
            A = Interval.Lopen(a0, a1)
            self.assertEqual(
                A.closure.interior.closure.interior,
                A.closure.interior,
            )
            self.assertEqual(
                A.interior.closure.interior.closure,
                A.interior.closure,
            )
            A = Interval.Lopen(a0, a1)
            self.assertEqual(
                A.closure.interior.closure.interior,
                A.closure.interior,
            )
            self.assertEqual(
                A.interior.closure.interior.closure,
                A.interior.closure,
            )
            A = Interval.Lopen(a0, a1)
            self.assertEqual(
                A.closure.interior.closure.interior,
                A.closure.interior,
            )
            self.assertEqual(
                A.interior.closure.interior.closure,
                A.interior.closure,
            )
            A = Interval.Ropen(a0, a1)
            self.assertEqual(
                A.closure.interior.closure.interior,
                A.closure.interior,
            )
            self.assertEqual(
                A.interior.closure.interior.closure,
                A.interior.closure,
            )
            A = Interval.Ropen(a0, a1)
            self.assertEqual(
                A.closure.interior.closure.interior,
                A.closure.interior,
            )
            self.assertEqual(
                A.interior.closure.interior.closure,
                A.interior.closure,
            )
            A = Interval.open(a0, a1)
            self.assertEqual(
                A.closure.interior.closure.interior,
                A.closure.interior,
            )
            self.assertEqual(
                A.interior.closure.interior.closure,
                A.interior.closure,
            )


if __name__ == "__main__":
    main()

入出力結果(Zsh、PowerShell、Terminal、Jupyter(IPython))

% ./sample10.py -v
10.
test_boundary_intersect_closure_union (__main__.Test) ... ok

----------------------------------------------------------------------
Ran 1 test in 1.508s

OK
%