# Thread: How to show Topology and find basis

1. ## How to show Topology and find basis

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

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

Show that the collection

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

is a topology on X.

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

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

3. Originally Posted by Plato
Please try to clarify how $\displaystyle \mathcal{T}$ is defined.

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

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

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

Intersection is the same idea.