# Thread: Prove continuity of a function (topological space)

1. ## Prove continuity of a function (topological space)

Hi!. I have problems with this proof

Let $f: (x ,t_x)\longrightarrow{(y,t_y)}$ , $B_x$ base of $t_x$ and $B_y$ base of $t_y$

I have to prove that

$f$ is continuous $\Longleftrightarrow{\forall{B\in{B_y}}}, \exists{B^{\prime}\in{B_x}} | B^{\prime}\subseteq{f^{-1}(B)}$

Continuous of functions (definition)

Let $X$ and $Y$ be topological spaces. A function $f: X\longrightarrow{Y}$ is said to be continuous if for each open subset $V$ of $Y$, the set $f^{-1}(V)$ is an open subset of X

2. ## Re: Prove continuity of a function (topological space)

Originally Posted by cristianoceli
Let $f: (x ,t_x)\longrightarrow{(y,t_y)}$ , $B_x$ base of $t_x$ and $B_y$ base of $t_y$
$f$ is continuous $\Longleftrightarrow{\forall{B\in{B_y}}}, \exists{B^{\prime}\in{B_x}} | B^{\prime}\subseteq{f^{-1}(B)}$
In a topological space $(X,\mathcal{T})$, the statement that $\mathscr{B}$ is a base for $\mathcal{T}$ means that $\mathscr{B}\subseteq\mathcal{T}$ so that $(\forall\mathcal{O}\in\mathcal{T})(\exists \mathcal {B}\in \mathscr {B})[\mathcal{B}\subset\mathcal{O}]$
It is easy to see that means every open set is the union of base elements.

Can you use the above to prove this?

3. ## Re: Prove continuity of a function (topological space)

Ok, but the other implication is a bit difficult

4. ## Re: Prove continuity of a function (topological space)

Originally Posted by cristianoceli
Ok, but the other implication is a bit difficult
By other are we to assume that you mean that if the inverse image of any basis set contains a basic set then the function is continuous?
Recall that the union of basis elements is an open set. We want to prove that the inverse image of every open set is an open set.
So can we show that ${f^{ - 1}}(O) = {f^{ - 1}}\left( {\bigcup\limits_\alpha {{B_\alpha }} } \right) = \bigcup\limits_\alpha {{f^{ - 1}}({B_\alpha })}~?$

P.S. Can you tell us the text book you are using?

5. ## Re: Prove continuity of a function (topological space)

I'm using the book Topology James. R. Munkres