Consider the definition of CW complex from wikipedia. It is assumed that the space is Hausdorff.
Are there problems if we drop this assumption? What is an example of a space satisfying all the CW complex axioms except this condition?
Consider the definition of CW complex from wikipedia. It is assumed that the space is Hausdorff.
Are there problems if we drop this assumption? What is an example of a space satisfying all the CW complex axioms except this condition?
Take two copies of $\mathbb{R}$, say $R_1$ and $R_2$. If $x\neq 0$ then identify $x\in R_1$ with $x\in R_2$. Then the two 0's will not have disjoint neighborhoods. This is also an example of a manifold that is not hausdorff.
Here's a proof that the space given by Joe Johnson is not Hausdorff:
Let $R_1= R_2=\mathbb{R}$ and let $X$ denote the quotient of $R_1\dot{\cup}R_2$ by the relation given by above. Let $\pi\colon R_1\dot{\cup}R_2\rightarrow X$ be the projection map, and let $\pi_1,\pi_2$ be the restrictions to the $R_1,R_2$ (note that restricting a continuous function gives a continuous function).
Let $U_1,U_2$ be any open nhds of $0_1$ and $0_2$ respectively. Since $\pi_1^{-1}(U_1)\subset\mathbb{R}$ is open and contains $0$, $\exists \epsilon_1>0\ :\ (-\epsilon_1,\epsilon_1)\subset \pi_1^{-1}(U_1)$. Similarly, $\exists \epsilon_2>0\ :\ (-\epsilon_2,\epsilon_2)\subset \pi_2^{-1}(U_2)$. Let $\epsilon=$min{${\epsilon_1,\epsilon_2}$}. Then $[\epsilon/2]=\pi_1(\epsilon/2)=\pi_2(\epsilon/2)\in U_1\cap U_2$