note-math

Subtopology := let $𝑆\subset 𝑋$. Inherit #link(<topology>)[net system] ${\text{T }}_{𝑆}=\left\{𝑆\cap 𝐵:𝐵\in {\text{ T}}_{𝑋}\right\}$

Equivalently defined, the subtopology is the smallest topology that makes the embedding map $𝑆\hookrightarrow 𝑋$ continuous

${𝑆}^{\prime }\subset 𝑆⟹\text{closed}\left({𝑆}^{\prime },{\text{T}}_{𝑆}\right)=𝑆\cap \text{closed}\left({𝑆}^{\prime },{\text{T}}_{𝑋}\right)$

Example $\text{closed}\left(\left(0,1\right],{\text{T}}_{\left(0,1\right]}\right)=\left(0,1\right]\ne \left[0,1\right]=\text{closed}\left(\left(0,1\right],{\text{T}}_{\mathrm{ℝ}}\right)$

Indicates that $\text{closed}\left({𝑆}^{\prime },{\text{T}}_{𝑋}\right)$ may have #link(<limit-point>)[limit point] $\notin 𝑆$ or $\in \text{closed}\left(𝑆,{\text{T}}_{𝑋}\right)\setminus 𝑆$, but the limit point of $\text{closed}\left({𝑆}^{\prime },{\text{T}}_{𝑆}\right)$ can only be $\in 𝑆$

$𝑆$ is a closed set

• ==> $𝑆=\text{closed}\left(𝑆,{\text{T}}_{𝑋}\right)\supset \text{closed}\left({𝑆}^{\prime },{\text{T}}_{𝑋}\right)$
• ==> $\text{closed}\left({𝑆}^{\prime },{\text{T}}_{𝑆}\right)=\text{closed}\left({𝑆}^{\prime },{\text{T}}_{𝑋}\right)$

${\text{T}}_{\frac{𝑋}{\sim }}$ := The largest topology that makes the quotient map $𝑋\twoheadrightarrow \frac{𝑋}{\sim }$ continuous, i.e. $𝐵\in {\text{ T }}_{\frac{𝑋}{\sim }}⟺{𝑓}^{-1}\left(𝐵\right)\in {\text{ T}}_{𝑋}$

${\text{T}}_{\prod _{𝑖\in 𝐼}{𝑋}_{𝑖}}$ := The smallest topology in which all component mappings ${𝑓}_{𝑖}:\prod _{𝑖\in 𝐼}{𝑋}_{𝑖}\to 𝑋$ are continuous, i.e., with the family of sets

$$\left\{{𝑓}_{𝑖}^{-1}\left(𝐵\right):\exists 𝑖\in 𝐼,\exists 𝐵\in {\text{ T}}_{𝑖}\right\}$$

to generate a net system of finite intersections

$${𝑓}_{{𝑖}_1}^{-1}\left({𝐵}_{{𝑖}_1}\right)\cap \cdots \cap {𝑓}_{{𝑖}_{𝑛}}^{-1}\left({𝐵}_{{𝑖}_{𝑛}}\right)$$

Because ${𝑓}_{𝑖}$ is a component mapping

$$\forall {𝐴}_{𝑗}\in \text{Subset}\left({𝑋}_{𝑗}\right),{𝑓}_{𝑖}\left({𝑓}_{𝑖}^{-1}\left({𝐴}_{𝑖}\right)\right)=\{\begin{array}{c}{𝐴}_{𝑖}\text{ if }𝑖=𝑗\\ {𝑋}_{𝑖}\text{ if }𝑖\ne 𝑗\end{array}$$

The topology of $\underset{𝑖\in 𝐼}{⨆}{𝑋}_{𝑖}$ is the largest topology that makes the embedding ${𝑋}_{𝑖}\hookrightarrow \underset{𝑖\in 𝐼}{⨆}{𝑋}_{𝑖}$ continuous

The point grid system ${\text{T}}_{𝑖}$ of ${𝑋}_{𝑖}$ for all $𝑖$ constitutes the point grid system of the sum space in the copy of the sum space, in which the form of the set is ${⨆}_{𝑖\in 𝐼}{𝐵}_{𝑖}$