Bases

Base of a topology

1. Base

   We say that $\mathcal{B}\subseteq P(X)$ is a base for a topology on $X$, if:

   • (B1): For all $x\in X$, there is $B\in \mathcal{B}$ such that $x\in B$.
   • (B2): If $B_{1}, B_{2}\in \mathcal{B}$ and $x\in B_{1}\cap B_2$, there is $B_{3}\in \mathcal{B}$ such that $x\in B_{3}\subseteq B_{1}\cap B_{2}$.

2. Generated topology

   Let $X$ be a set and $\mathcal{B}$ be a base for a topology on $X$. Then

   $$\tau_{\mathcal{B}}=\{U\subseteq X\mid \forall x\in U\,\exists B\in \mathcal{B} \text{ such that } x\in B\subseteq U\}$$

   is a topology on $X$.

Proof of generated topology

1. Proof
   • That $\emptyset\in\tau_{\mathcal{B}}$ is vacuously true. $X\in\tau_{\mathcal{B}}$ because of (B1).
   • Let $U_{\alpha}\in\tau_{\mathcal{B}}$ for all $\alpha\in I$, and let $x\in \cup U_{\alpha}$.

   • Assume $x\in U_{\alpha_{0}}$. Then there is $B\in \mathcal{B}$ such that:

     $$x\in B\subseteq U_{\alpha_{0}}\subseteq \cup U_{\alpha},$$

     which proves (T2).

   • Now, let $U_{1},U_{2}\in\tau_{\mathcal{B}}$, and let $x\in U_{1}\cap U_{2}$. Let $x\in B_{i}\subseteq U_{i}$ for $i=1,2$.
   • Using (B2), find $B_{3}\in \mathcal{B}$ such that $x\in B_{3}\subseteq B_{1}\cap B_{2}$.

   • Then

     $$x\in B_{3}\subseteq B_{1}\cap B_{2}\subseteq U_{1}\cap U_{2},$$

     which proves (T3). $\Box$

Union of base elements

1. Remark

   Note that all elements of $\mathcal{B}$ are open sets in $\tau_{\mathcal{B}}$.

2. Elements of generated topology

   Let $X$ be a set and $\mathcal{B}$ a base for a topology. Then $\tau_{\mathcal{B}}$ is equal to the collection of subsets of $X$ that are union of elements of $\mathcal{B}$.

3. Proof

   • If $U$ is union of elements of $\mathcal{B}$, then, since $\mathcal{B}\subseteq\tau_{\mathcal{B}}$, and $\tau_{\mathcal{B}}$ is closed under arbitrary unions, then $U\in\tau_{\mathcal{B}}$.
   • Conversely, if $U\in\tau_{\mathcal{B}}$, let $x\in U$. By definition of $\tau_{\mathcal{B}}$, there is $B_{x}\in \mathcal{B}$ such that $x\in B_{x}\subseteq U$. Hence $U=\cup_{x\in U}B_{x}$. $\Box$

Exercises

1. Show that if $\mathcal{B}$ is a base for a topology on $X$, then $\tau_{\mathcal{B}}$ is equal to the intersection of the topologies on $X$ that contain $\mathcal{B}$.

2. Show that

   $$\{(a,b)\mid a,b\in \mathbb{Q}, a<b\}$$

   is a basis for the usual topology on $\mathbb{R}$.

3. Show that

   $$\{[a,b)\mid a,b\in \mathbb{R}, a<b\}$$

   is a basis for a topology that is different from the usual topology on $\mathbb{R}$.

4. Let $\mathcal{B}$ be a basis for a topology, and let $\mathcal{C}\subseteq P(X)$ be such that $\mathcal{B}\subseteq \mathcal{C}$. Is $\mathcal{C}$ a basis for a topology?
5. Let $\mathcal{B}$ be a basis for a topology on $X$ such that $\tau_{\mathcal{B}}$ is the discrete topology. If $x\in X$, show that $\{x\}\in \mathcal{B}$.

Links

• Base (topology) - Wikipedia, the free encyclopedia