Definition 3.1.4: Base of Topology

Definition 3.1.4 (Base of Topology).label Let $X$ be a set then $\cali{B}\suf\cali{P}(X)$ is a base if it satisfies:

(B1)
For each $x\in X$ there exists $B_{x}\in\cali{B}$ such that $x\in B_{x}$
(B2)
If $x\in B_{1},B_{2}\in\cali{B}$ then there exists $B_{3}\in\cali{B}$ such that $x\in B_{3}\suf B_{1}\cap B_{2}$

then the topology generated by $\cali{B}$ defined as

\begin{align*}\tau(\cali{B})\define\curl{\cups{}{i\in I}U_i:\curl{U_i}_{i\in I}\suf \cali{B},I\text{ index set}}\end{align*}

is a topology on $X$. Suppose $X$ has a topology $\tau$ then we say $\cali{B}$ is a base for $\tau$ if $\cali{B}\suf \tau$ and

(B’1)
$\cali{B}$ satisfies (B1)
(B’2)
For each $x\in X, U\in\tau$ such that $x\in U$, there exists $B\in\cali{B}$ such that $x\in B\suf U$

in which case $\tau=\tau(\cali{B})$.

Proof. If $\cali{B}$ is a base then $\emp=\cups{}{\emp}\in\tau(\cali{B})$ and by (B1) $X=\cups{}{x\in X}B_{x}\in \tau(\cali{B})$ showing (O1). Observe that for any $\curl{U_i}_{i\in I},\curl{V_j}_{j\in J}\suf\cali{B}$ and $*\in \curl{\cap,\cup}$ we have

\begin{align*}\parens{\cups{}{i\in I}U_i}*\parens{\cups{}{j\in J }V_j}=\cups{}{i\in I}\cups{}{j\in J}U_{i}* V_{j}\end{align*}

which shows (O2) and by (B2) this observation also shows (O3).

Suppose now that $\cali{B}$ is a base for $\tau$ then by (O3) and $\cali{B}\suf \tau$ we have $\tau(\cali{B})\suf \tau$. For any $U\in \tau$ either $U=\emp\in\tau(\cali{B})$, or $U\neq \emp$. In which case by (B’2) for each $x\in U$ pick $B_{x}\in\cali{B}$ such that $x\in B_{x}\suf U$ then $U=\cups{}{x\in U }B_{x}\in \tau(\cali{B})$ as desired.$\square$

The Eigenspace

The Eigenspace