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 $(X, \mathfrak{T})$. Then for arbitrary subset $A \subseteq X$, we have $$ \mathfrak{T}_A = \{A \cap U | U \in \mathfrak{T}\} $$ is a topology on A. This is said to be the subspace topology.
   If $U \in \mathfrak{T}_A$, we say $U$ is open in $A$.

2. Suppose we have a topology $\mathfrak{T}$ generated by a basis $\mathfrak{B}$, then $$ \mathfrak{B}_Y = \{B \cap Y | B \in \mathfrak{B}\} $$ is a basis for the subspace topology on $Y$.

Lemma 2. Suppose $Y$ is a subspace of $X$. If $U$ is open in $Y$ and $Y$ is open in $X$, we have $U$ is open in $X$.

References

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