General Topology - Subspace Topology

I hate topology. --- By Woziji Shuode

RECAP

Please read to "General Topology - Topology and Basis".

Definition

Definition 1. [Subspace Topology]¹

1. Suppose we have a topology space . Then for arbitrary subset , we have is a topology on A. This is said to be the subspace topology.
   If , we say is open in .

2. Suppose we have a topology generated by a basis , then is a basis for the subspace topology on .

Lemma 2. Suppose is a subspace of . If is open in and is open in , we have is open in .

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1. Prof. Ruobing Zhang's lecture note of MATH551, UW-Madison↩︎