Problem 5 part d), of chapter V, section 3 (p. 118):
d. In $E^2$ let $L_1$ be the line $x=1$ and $L_2$ the line $x=-1$. For each $n\in\Bbb Z^+$ let $R_n$ be the rectangle $\{(x,y):|x|\le \frac{n}{n+1},|y|\le n\}$. Finally, let $Y$ be the subspace $L_1\cup L_2\cup \bigcup_{n\in\Bbb Z^+} Fr(R_n)$ of $E^2$. Show: The component of $(1,0)$ is $L_1$ and the quasi-component of $(1,0)$ is $L_1\cup L_2$.
I can see why the component of $(1,0)$ is $L_1$, that's just using the definition. However I can't see why the quasicomponent is $L_1\cup L_2$, well intuitively it makes sense, because when $n\to\infty$, the rectangle $R_n$ looks like (or may I say, exactly like) the stripe formed by $L_1\cup L_2$, do I have to show explicitly that $L_1\cup L_2 \subseteq \bigcup_{n\in\Bbb Z^+} Fr(R_n)$? I don't know this makes sense at times.
Definition of quasi-component:
In a space $X$, define $x\sim y$ if there is no decomposition of $X$ into two disjoint open sets, one of which contains $x$ and the other $y$. A quasi-component is the equivalent class of such relation.
I take as follows, if $Q_X(x)$ is a quasi component, then $Q_X(x)=\{y: \text{if $x,y\in A$ then $A$ is a connected set}\}$
(I'll use $\partial$ instead of $\text{Fr}$ if you don't mind, I'm just more used to it.)
You have to show that for $x=(1,0)$ and $y\in L_2$ there is no separation of $X$ between $x$ and $y$. A separation would amount to a clopen set containing $x$ (and thus $L_1$) but not containing $y$. If $S$ is such a clopen subset of $X$ with $x\in S$, can you show that it intersects arbitrarily large $∂R_n$? Can you conclude that $S$ must contain $L_2$, using the fact that $∂R_n$ is connected for each $n$?
Don't forget that you also have to show that for a point $z\in X\setminus(L_1\cup L_2$), there does exist a separation of $X$ between $x$ and $z$.