If $f$ is a continuous function from $X$ to $Y$ and $A$ is a subset of $X$ then is it true that $x \in A^d$ implies $f(x) \in (f(A))^d$? Here $A^d$ is the derived set of $A$.
If $f$ is a continuous function from $X$ to $Y$ and $X$ is a compact space,then show that $f$ is bounded and attains its bounds.
$A$ and $B$ are two compact subsets of a Hausdorff space $Y$.
i. Union of $A$ and $B$ is compact in $Y$:
$A$ and $B$ are closed and hence the union is closed,so it is compact.
ii. Intersection of $A$ and $B$ is compact in $Y$:
$A$ and $B$ are closed and hence the intersection is closed,so it is compact.
iii. $\operatorname{fr}(A)$ is compact:
$\operatorname{fr}(A)$ is the intersection of $\operatorname{cl}(A)$ and $\operatorname{cl}(Y-A)$. Intersection of two closed sets, hence compact. Am I correct?
Topology-Continuous functions and compact spaces
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general-topology
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0What's the derived set of a set? – 2012-12-26
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0set of limit points of that set. – 2012-12-26