Prove that A compact hausdorff space is generated by a weak topology of C(X,R) Prove that A compact hausdorff space is a weak topology generated by 
C(X,R) ,by using C(X,R) separates points of X i.e for x not equal to y there exists a function f in C(X,R) s.t f(x) not equal to f(y), Where 
C(X,R)is the set of all continuous real valued function from X to R.
 A: I am still perplexed by this question and I think you are a bit confused about what various words mean and how to use them.  But it looks like the question you were trying to ask is:

Let $X$ be a set and let $\tau$ be a compact Hausdorff topology on $X$.  Prove that $\tau$ coincides with the weak topology on $X$ induced by the set of all $\tau$-continuous functions $X \to \mathbb{R}$.

If this is the correct interpretation of your question, then we can use the universal mapping property of the weak topology: the weak topology on $X$ induced by $C(X,\mathbb{R})$ is the unique topology with the property that a function $g \colon Z \to X$ (where $Z$ is any topological space) is continuous if and only if $f \circ g \colon Z \to \mathbb{R}$ is continuous for every $f \in C(X,\mathbb{R})$.  By uniqueness, we simply need to show that the original topology $\tau$ has this property.
If $g \colon Z \to X$ is continous then $f \circ g \colon Z \to \mathbb{R}$ is clearly $\tau$-continuous for any $\tau$-continuous function $f \colon X \to \mathbb{R}$.  Conversely, suppose $f \circ g \colon Z \to \mathbb{R}$ is continuous for every $\tau$-continuous function $f$; we want to prove that $g$ is $\tau$-continuous.  Let $z \in Z$ be any point and let $U \subseteq X$ be a neighborhood of $g(z)$.  By Urysohn's lemma there is a function $f \colon X \to [0,\infty)$ such that $f(g(z)) = 0$ and $f(x) = 1$ for every $x \in X - U$, so that $f^{-1}([0,1)) = U$.  But then 
$$g^{-1}(U) = g^{-1}(f^{-1}([0,1))) = (f \circ g)^{-1}([0,1))$$
and this is an open set containing $z$ since $f \circ g$ is by assumption continuous.  Thus $g$ is continuous.
A: I think I have got 1 answer let C(X,R) be set of all functions by which weak topology is generated. Now C(X,R) separates points, so X is hausdorff & by completely regularity I have compactness. 
