Quotient Topology

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最新推荐文章于 2025-02-06 16:32:39 发布

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分类专栏： math 文章标签： topology

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Quotient Topology

notations

top: topological space (structure)
hom: homomorphism on top

1. Definition

Quotient Space 1

Suppose

X $X$ is a top,

f:X→Y $f:X\to Y$ . Quotient topology of

Y $Y$ is

{B⊂Y|f−1[B]∈X} $\{B\subset Y | f^{-1}[B]\in \mathcal{T}_X\}$ denoted as

X/f $X/f$ . Under this topology

Y $Y$ is the quotient space of

X $X$ .

Quotient topology is the finest top for $Y$ that $f$ is continous, meanwhile $f$ is a quotient map. Remember $f^{-1}[A]\in \mathcal{T}_X \iff A \in \mathcal{T}_{X/f}$

Suppose $X$ is a top, $\sim$ is an equivalent relation (or partition), let $f(x)=[x]$ (equivalent class or component). Quotient space of $X$ under $\sim$ is $X/f=\{A | \bigcup_{x\in A}[x]\}$ denoted as $X/\sim$ .; A partition $\pi$ gives an equivalent relation $x\sim y$ iff $x,y\in A\in\pi$ . Define $X/\pi=X/\sim$; if $f$ is open, then $f$ is a quotient map, not the other way. In topological group, the quotient map is indeed open.; $A\subset X$ , given a partition $\pi=\{A,\{x\},\{y\},\cdots\}$ , define gluing space $X/A=X/\pi$ . Element $\{x\}$ in $X/A$ is denoted as $x$ for convenience.
To glue two spaces: $X\cap Y=\emptyset$ , $x\in X,y\in Y$ , then $X+_{x,y}Y=X\cup Y/\{x,y\}$ .