1. 教科書

Topology without tears

位相空間の学び直しに使用したネットに落ちてる教科書.説明が凄く丁寧なので初学者にもオススメ.事前知識は集合論のみ必要.実数の構成とか濃度の概念とか知ってるとつまずくところはないかも.

2. 証明の補足

• second contable base contains a countable base, hence separable
• uncountable set with cofinite topology is not 1st countable
• Dense subset of $R$ with the Eucliean topology is dense in Sorgenfrey line. https://etd.auburn.edu/bitstream/handle/10415/393/GREIWE_REGINA_16.pdf?sequence=1
• Sorgenfrey line is not second countable
• Lindelofspace Every uncountable subset of R contains one of its limit points for Euclidean topology
• Exercises4.3-#8-(ⅰ)
• Exercises4.3-#8-(Ⅱ)
• Exercises4.3-#8-(Ⅲ)
• Exercises4.3-#8-(IV),(V)
• Any subspace of separable metric space is seperable
• Open subset of Polish space is Poli space (Lemma3.4)
• A subset of Polish Space is Polish if and only if it is a G_\delta set there. P6 > Theorem 7