Following task in a textbook:
Give a characterization of the subspace topology through:
a) closed sets
b) closure
c) neighborhoods
And I am not sure what actually is asked here. So what a suitable characterization should fulfill. Of course it should be equivalent.
So let $(X,\tau)$ be a topological space, and $(Y,\tau_Y)$ a subspace with the subspace topology. For a) I thought about this at first. Take $\{A\subseteq Y: A=Y\cap U^c~\text{with $U\in\tau$}\}$.
As $U^c$ is closed, this (somewhat) gives a characterization with closed sets. But this was not satisfying at all.
It is easy to show that a set $A\subseteq Y$ is closed (in $Y$) iff it exists a closed set (in $X$) $W\subseteq X$ with $A=Y\cap W$.
Which seems like a more suitable answer, and I could discribe the subspace topology in terms of closed sets in the obvious way.
But what about b)?
I know that if $A\subseteq Y$ then $\overline{A}\cap Y$ is the closure of $A$ in $Y$. But how would that lead to a characterization of the subspace topology?
It feels more like I am giving a characterization of the terms closed set, closure, neighborhood with regards to the subspace topology.
So I would say that $U\subseteq Y$ is a neighborhood of $y\in Y$ (with regards to the subspace topology), iff there is a neighborhood $V\subseteq X$ of $y$ with $U=Y\cap V$.
But I am not sure about the solution the task wants you to find. Can you help me out. Are my thoughts correct?
Thanks in advance.