The first one: $$\begin{array}{rcl} y &=& y' \ln y' \\ W(y) &=& \ln y' \\ y' &=& e^{W(y)} \\ \dfrac{\mathrm dy}{\mathrm dx} &=& e^{W(y)} \\ \dfrac{\mathrm dx}{\mathrm dy} &=& e^{-W(y)} \\ x &=& \displaystyle \int e^{-W(y)} \ \mathrm dy \\ &=& \displaystyle \int e^{-u} \ \mathrm d(ue^u) \\ &=& \di...

Let $A=\{0 \} \cup \{\frac {1}{n}|n=1,2,3, . . . \}$ be a subspace of $E^1$ My question : (a) is the singleton set $\{\frac {1}{n}\}$ open in $A$ ? For all $n \in N$, the singleton $\{\frac {1}{n}\}$ is a closed subset of $E^1$. Let $X=\cup \{\frac {1}{n}\} $. Then, since the sequence $\{...