# Math Help - clousre

1. ## clousre

Suppose that $\displaystyle \tau 1 \mbox and \quad \tau2 \mbox \quad are \quad \quad \quad topologies \quad on \quad a set X \quad with \quad \tau1 \subset \tau2 . \mbox \quad if A \subset X \quad$

$\mbox prove \quad that : \quad \overline{A}^2 \subset \overline {A}^1 \mbox \quad where \overline {A}^1 \mbox is \quad the \quad clousre$ $\quad of A \quad in (X, \tau1 ) \quad \overline {A}^2 \mbox is \quad the \quad clousre \quad of A \quad in (X,\tau2)$

2. If $A$ is an open subset in $\tau_1$ then $A$ is open in $\tau_2$. So, if $B$ is closed in $\tau_1$ then $B$ is closed in $\tau_2$. Now you define the closure of $A$ in $\tau_i$ as the intersection of all closed subsets of $(X, \tau_i)$ such that they contain $A$, but since closed sets in $\tau_1$ are closed in $\tau_2$ then the closure in $\tau_2$ has more elements to intersect and thus is contained in the closure in $\tau_1$.

Well, that is an overview of the proof. I leave the details to you.

3. Here is a different approach.
$x \in \overline A$ if and only if each for each open set $O$, $x \in O\, \Rightarrow \,O \cap A \ne \emptyset$.
But $O \in \tau _1 \; \Rightarrow \;O \in \tau _2$.