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**Connectedness and "fineness" of topology**

## Homework Statement

Let T and T' be two topologies on X, with T' finer than T. What does connectedness of X in one topology imply about connectedness in the other?

## The Attempt at a Solution

Assume (X, T) is connected, so there don't exist two disjoint, open and non-empty sets U, V whose union is X. Open sets U and V in T can be written as unions of basis elements Bx and By, where x are elements from U and y from V. Since T' is finer than T, U and V can as well be written as unions of basis elements B'x and B'y from T', where x and y are elements in U and V, respectively (since for any element x of U and Bx of T containing x there exists a basis element B'x of T' which contains x and is contained in Bx). So, connectedness of X in T implies connectedness of X in T', right?

Thanks for any replies, I hope I was clear enough, too lazy to TeX. :)