How to show Topology and find basis

**flaming** · April 8th 2009, 09:42 AM

1 (a)
Let $\mathcal{C}$ be a collection of subset of the set X, satisfying:

i. $\phi$ and X contained in $\mathcal{C}$ ,
ii. finite unions of elements in $\mathcal{C}$ are in $\mathcal{C}$ , and
iii. arbitrary unions of elements in $\mathcal{C}$ are in [ $\mathcal{C}$

Show that the collection

$\mathcal{T} = \{X - C|C \in \mathcal{C} \}$

is a topology on X.

(b)
Give an example of such a topology when X = $\Re$ . What are the open sets in this topology?What are the closed sets? What is the basis for the topology?

**Plato** · April 8th 2009, 09:59 AM

Please try to clarify how $\mathcal{T}$ is defined.

**r2dee6** · April 8th 2009, 10:55 AM

Originally Posted by Plato

Please try to clarify how $\mathcal{T}$ is defined.

That is how T is defined in the question.I am not sure what you mean by the question..sorry

**Plato** · April 8th 2009, 12:18 PM

To aid in notation use this. $A \in T\; \Rightarrow \;\left( {\exists A' \in C} \right)\left[ {A = X\backslash A'} \right]$ .
Because $\emptyset = X\backslash X\;\& \;X = X\backslash \emptyset$ the first two requirements are fulfilled.

For union $\left\{ {A_\alpha } \right\}$ is a collection in $T$ .
$\bigcup {\left\{ {A_\alpha } \right\}} = \bigcup {X\backslash A'_\alpha } = X\backslash \bigcap {A'_\alpha }$ .

Intersection is the same idea.

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