Notes:Fundamentals of Algebraic Topology - Weintraub

TODO: Make up for the rush this was created in

Chapter 1

Section 1.1: Background

A pair $(X,A)$ consists of a space $X$ and a subspace $A\in\mathcal{P}(X)$. We let $f:X\rightarrow Y$ denote a map from $X$ to $Y$ as usual.

Similarly we let $f:(X,A)\rightarrow(Y,B)$ denote a map from the pair $(X,A)$ to $(Y,B)$, i.e. a map $f:X\rightarrow Y$ whose restriction is $f\big\vert_A:A\rightarrow B$; or more simply, a map $f:X\rightarrow Y$ with $f(A)\subseteq B$

For most purposes we can identify the pair $(X,\emptyset)$ with $X$. Then $f:X\rightarrow Y$ is a special case of $f:(X,\emptyset)\rightarrow (Y,\emptyset)$. The author notes this undermines clarity. Ha.

A homeomorphism, $f:X\rightarrow Y$ is a continuous map with a continuous inverse map, $g:Y\rightarrow X$. A homeomorphism $f:(X,A)\righarrow(Y,B)$ is defined similarly.

• I think this means $f:(X,A)\rightarrow (Y,B)$ with inverse $g:(Y,B)\rightarrow (X,A)$, this ends up with $f(A)=B$

The spaces or pairs are said to be homeomorphic.

This author uses $\approx$ for homeomorphic spaces (or "pairs", $(X,A)\approx(Y,B)$.

For a pair $(X,Y)$ we let $X/Y$ denote the quotient space.

We also have the homeomorphism:

• $f:\frac{[0,1]}{(\{0\}\cup\{1\})}\rightarrow\mathbb{S}^1$ by $f:t\mapsto e^{2\pi jt}$ which he does often use in the next sections.

Given two spaces $X$ and $Y$ we write $X\times Y$ for their product, as usual. We define the product of pairs by:

• $(X,A)\times(Y,B):=\big(X\times Y, (X\times B)\cup(A\times Y)\big)$