I have just started learning metric space and our professor has defined two terminologies:
$1.$Base of a topological space $(X,\tau)$ is defined to be a set $\scr B\subset \tau$ such that for each $U\in \tau$ and $x\in U$, $\exists B\in \scr B$ such that $x \in B \subset U$.
$2.$Local base at a point $x\in X$ in a topological space is a collection of neighbourhoods $(v_\alpha)_{\alpha \in \lambda} $of $x$ such that for any nbd $U_x$ of $x$,some $v_\alpha \subset U_x$.
Now I have some question(follow these definitions) regarding
what are the connections between local base and a global base?I need some properties that would help me to understand the connection between these two things and also enable me to work freely with these.Can somebody please help me with them?
Since I have yet not learnt topology,I do not know much of it.
Best Answer
Let be $(X,\mathcal{T})$ a topological space: first we observe that the collection $$ \mathfrak{B}=\{\mathcal{B}\subseteq\mathcal{T}:\text{ basis for }\mathcal{T}\} $$ of the basis $\mathcal{B}$ for $\mathcal{T}$ is not empty because trivially $\mathcal{T}\in\mathfrak{B}$; analogously for any point $x$ of $X$ the collection $$ \mathcal{V}(x)=\{V_x\subseteq X: V_x\text{ is a neighborhood of } x\} $$ of the neighborhood of $x$ is not empty because trivially $X\in\mathcal{V}(x)$ so the collecion $$ \mathfrak{V}(x)=\{\mathcal{B}(x)\subseteq\mathcal{P}(X):\mathcal{B}(x)\quad\text{is a local basis for } x\} $$ is not empty because trivially $\mathcal{V}(x)\in\mathfrak{V}(x)$.
Well it is not complicated to prove that for any $\mathcal{B}\in\mathfrak{B}$ and for any $x\in X$ the collection $$\mathcal{B}(x)=\{B\in\mathcal{B}:x\in B\} $$ is a local basis for $x$; analogously it is not complicated to prove that for any $x\in X$ the collection $$ \bigcup_{x\in X}\mathfrak{A}(x) $$
is a basis for $\mathcal T$ when $\mathfrak A(x)$ is the set $$ \mathfrak A(x):=\{\mathcal A(x)\in\mathfrak V(x):\mathcal A(x)\text{ is an open local base for }x\} $$ for all $x\in X$.