Can a set be closed in one topology but neither open nor closed in another? Can a set be closed in one topology but neither open nor closed in another?
If we say that the complement of a open set is a closed set, i.e. if $S \subseteq X$ is open then $X \setminus S$ is closed, I argue that depending on the topology, it could actually be the case. Consider the Euclidean topology on $\mathbb{R}$. Clearly any closed interval $[a,b]$ with $a,b \in \mathbb{R}$ is a closed set in this topology.
However, given the trivial topology on $\mathbb{R}$, any interval $[a,b]$ is neither closed or open.
But according to this Wikipedia page, an equivalent definition of a closed set is that "a set is closed if and only if it contains all of its limit points". And from this point of view, the inverval $[a,b]$ is clearly closed.
So i guess that my question is: How can two (supposedly) equivalent definitios of a closed sets, have a set be closed in one and not the other?

in discrete topology any set is closed as well as open... so $\mathbb{R}$ with discrete topology $[0,1)$ is open as well as closed but with resprct to euclidean topology $[0,1)$ nither closed nor open 
