I was given that question in HW1 in topology:
For all three real number $r,a,b$ s.t. $a<b$ we define: $u_{r,a,b}=\{f\in\mathbb{R}^{\mathbb{R}}\mid a<f(r)<b\}$
First prove that the collections of all finite intersections from the collections of the set:
$\{u_{r,a,b}\mid r,a,b\in\mathbb{R},\,a<b\}$
we defined is a basis for some topology $\mathcal{T}$ over $\mathbb{R}^{\mathbb{R}}.$
I did that. After that I need to decide if $(\mathbb{R}^{\mathbb{R}},\mathcal{T})$ satisfies the first countable axiom.
- Definition for Local base:
Let $(X,\mathcal{T})$ be a topological space and let $x \in X$. A Local Base of the element $x$ is a collection of open neighbourhoods of $x$, $B_x$ such that for all $U \in \mathcal{T}$ with $x \in U$ there exists a $B \in B_x$ such that $x \in B \subseteq U$
- Definition for first countable axiom:
Let $(X,\mathcal{T})$ be a topological space. Then $(X,\mathcal{T})$ is said to be a First Countable topological space if every point $x\in X$ has a countable local basis.
My idea (which is not formal at all):
I think it's not satisfy the first countable axiom because we need to take care of every single point (from the domain) separately, because we can take $f(x) = x$ and for all $r \in \mathbb{R}$ and for all $\varepsilon >0$ the open set $u_{r,r-\varepsilon,r+\varepsilon}$ and for every $r'\in \mathbb{R}$ s.t. $r' \neq r$ we can "separate" the open sets $u_{r,r-\varepsilon,r+\varepsilon}$ and $u_{r',r'-\varepsilon,r'+\varepsilon}$ so if one of the basis elements will be contained in one of them then it couldn't be contained in the other one, so we need at least $2^{\aleph_0}$ Basis elements, We can't take for example only rationals values because we won't cover the irrationals because what I said.
I assume the way to prove that is to assume in contradiction that for $f \in \mathbb{R}^\mathbb{R}$ I mentioned there is a countable local basis and from there apply my idea and we get a contradiction, But I can't figure out how to apply that idea formally.
I'll be glad if someone can help me to apply my idea formally or just give me a way to decide and prove the result.
I very baffled from that question so I sorry if my explanations aren't correct or difficult to understand, Sorry if I did grammar mistakes.
Thank you in advance!
