Subspace Topology

Definition

Given a topological space $(X,\mathcal{T})$ with a basis $\mathcal{B}$, and a subset of the set $Y\subset X$. The subspace topology on $Y$ and its basis is defined as ${\mathcal{T}}_Y=\{U\cap Y|U\in \mathcal{T}\},{\mathcal{B}}_Y=\{B\cap Y|B\in \mathcal{B}\}$

Facts

Consider a Subspace Topology $(Y,{\mathcal{T}}_Y)$ of a Topological Space $(X,\mathcal{T})$ and a subset $A\subset Y$. Then, $(Y\setminus A)\in {\mathcal{T}}_Y⟺(\exists (X\setminus C)\in \mathcal{T}\text{s.t.}A=C\cap Y)$

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Consider a Topological Space $(X,\mathcal{T})$ and a subset $A\subset X$. Then, $(X\setminus A)\in \mathcal{T}⟺A^{\prime }\subset A$ where $A^{\prime }$ is the set of all limit points of $A$

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Consider a subspace topology $(Y,{\mathcal{T}}_Y)$ of a Topological Space $(X,\mathcal{T})$. Then, $(U\in {\mathcal{T}}_Y\text{and}Y\in \mathcal{T})\Rightarrow U\in \mathcal{T}$

Consider two subspaces $A,B$ of the sets $X,Y$. The Product Topology on $A\times B$ is the subspace topology of $X\times Y$.

Product Topology $\to$ Subspace topology: $(U\times V)\cap (A\times B)$ Subspace topologies $\to$ Product Topology: $(U\cap A)\times (V\cap B)$ $(U\times V)\cap (A\times B)=(U\cap A)\times (V\cap B)$ where $U\in {\mathcal{T}}_X,V\in {\mathcal{T}}_Y$.