Local Bases of a Point in a Topological Space

Local Bases of a Point in a Topological Space

Recall from the Bases of a Topology page that if $(X, \tau)$ is a topological space then a basis $\mathcal B$ of $\tau$ is a collection of subsets from $\tau$ such that each $U \in \tau$ is the union of some subcollection $\mathcal B^* \subseteq \mathcal B$ of $\mathcal B$, i.e., for all $U \in \tau$ we have that there exists a $\mathcal B^* \subseteq \mathcal B$ such that:

(1)
\begin{align} \quad U = \bigcup_{B \in \mathcal B^*} B \end{align}

We will now look at a similar definition called a local bases of a point $x$ in a topological space $(X, \tau)$.

Definition: Let $(X, \tau)$ be a topological space and let $x \in X$. A Local Basis of the element $x$ is a collection of open neighbourhoods of $x$, $\mathcal B_x$ such that for all $U \in \tau$ with $x \in U$ there exists a $B \in \mathcal B_x$ such that $x \in B \subseteq U$.

In other words, a local basis of the point $x \in X$ is a collection of sets $\mathcal B_x$ such that in every open neighbourhood of $x$ there exists a basis element $B \in \mathcal B_x$ contained in this open neighbourhood.

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For example, consider the topological space $(\mathbb{R}, \tau)$ where $\tau$ is the usual topology of open intervals on $\mathbb{R}$. Consider the point $0 \in \mathbb{R}$. One such local basis of $0$ is the following collection:

(2)
\begin{align} \quad \mathcal B_0 = \{ (a, b) : a, b \in \mathbb{R}, a < 0 < b \} \end{align}

For example, if we consider the open set $U = (-1, 1) \cup (2, 3) \in \tau$ which contains $0$, then for $B = \left ( - \frac{1}{2}, \frac{1}{2} \right ) \in \mathcal B_0$ we see that $0 \in B \subseteq U$.

More generally, for any $x \in \mathbb{R}$, a local basis of $x$ is

(3)
\begin{align} \quad \mathcal B_x = \{ (a, b) : a, b \in \mathbb{R}, a < x < b \} \end{align}

This is because for any open set $U \in \tau$ containing $x$ there will be an open interval containing $x$ that is contained in $U$.

For a different example, consider the set $X = \{ a, b, c, d, e \}$ and the topology $\tau = \{ \emptyset, \{a \}, \{a, b \}, \{a, c \}, \{a, b, c \}, \{a, b, c, d \}, X \}$.

What is a local basis for the element $b \in X$? Let's first look at the sets in $\tau$ containing $b$. They are $U_1 = \{ a, b \}$, $U_2 = \{ a, b, c \}$, $U_3 = \{a, b, c, d \}$, and $U_4 = X$.

We see that $\mathcal B_b = \{ \{ b \} \}$ works as a local basis of $b$ since:

(4)
\begin{align} \quad b \in \{ b \} \subseteq U_1 = \{a, b \} \quad b \in \{ b \} \subseteq U_2 = \{a, b, c \} \quad b \in \{ b \} \subseteq U_3 = \{a, b, c, d \} \quad b \in \{ b \} \subseteq U_4 = X \end{align}

What is a local basis for the element $c \in X$? The sets in $\tau$ containing $c$ are $U_1 = \{a, c \}$, $U_2 = \{a, b, c \}$, $U_3 = \{ a, b, c, d \}$, and $U_4 = X$.

We see that $\mathcal B_c = \{ \{ a, c \} \}$ works as a local basis of $c$ since:

(5)
\begin{align} \quad c \in \{ a, c\} \subseteq U_1 = \{a, c \} \quad c \in \{a, c \} \subseteq \{a, b, c \} \quad c \in \{a, c \} \subseteq \{a, b, c, d \} \quad c \in \{a, c\} \subseteq X \end{align}
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