I am studying some topology for my thesis (I come from a different area of mathematics) and I have some basic questions which I have difficulty to clear out.
I would really appreciate some help (also a good reference book if possible).
Suppose $X \subseteq Y$ equipped with the topologies $A$ and $B$ respectively, where the topology $A$ is stronger. (Also $X$ is not metrizable, but $Y$ is. I don't know if this helps but I am writing it in any case)
I would like to know which of the following statements are true and which are false :
1) $A$ topology being stronger means that convergence wrt $A$ $\implies$ convergence wrt $B$.
2) $A\supseteq B$.
3) An open set in $X$ is not necessarily open in $Y$ (since if the previous question is correct then it can be in A but not in B).
4) the Borel $\sigma$-field defined by $A$ contains the Borel $\sigma$-field defined by $B$.
I'll call the topologies $\mathcal{T}_1$ and $\mathcal{T}_2$ respectively, both defined on $Y$, and suppose $\mathcal{T}_2$ is stronger than $\mathcal{T}_1$.
This is by definition that the inclusion $\mathcal{T}_1\subseteq \mathcal{T}_2$ holds. I.e. there can be open sets in $(Y,\mathcal{T}_2)$ that are not open in $(Y, \mathcal{T}_1)$; I fail to see the relevance of your $X$, except that $X \subseteq Y$ can be $\mathcal{T}_2$-open and not $\mathcal{T}_1$-open. These are just reformulations of course.
If $y_n \to y$ in $\mathcal{T}_2$ (the stronger one) then certainly $y_n \to y$ in $\mathcal{T}_1$ as well. The inclusion-definition immediately
The Borel $\sigma$-algebra genererated by $\mathcal{T}_1$ will be contained in that generated by $\mathcal{T}_2$, which is obvious from the inclusion-definition too.