Bases of Topo., Local Bases of Points, and Subbases of Topo. Review

Bases of Topologies, Local Bases of Points, and Subbases of Topologies Review

We will now review some of the important information regarding bases of topologies, local bases of points, and subbases of topologies.

Let $(X, \tau)$ be a topological space.

  • Recall from the Bases of a Topology page that a Basis for the topology $\tau$ is a collection $\mathcal B$ of open sets ($\mathcal B \subseteq \mathcal \tau$) such that every $U \in \tau$ can be expressed as the union of some subcollection of sets from $\mathcal B$. In other words, for each $U \in \tau$ there exists a $\mathcal B^* \subseteq \mathcal B$ such that:
(1)
\begin{align} \quad U = \bigcup_{B \in \mathcal B^*} B \end{align}
  • The most trivial basis for the topology $\tau$ is $\tau$ itself.
(2)
\begin{align} \quad x \in B \subseteq U \end{align}
  • Afterwards, on the Generating Topologies from a Collection of Subsets of a Set page we looked at a very important theorem that says a collection of subsets of $X$, $\mathcal B$, is a basis for "SOME" topology $\tau$ on $X$ if and only if $X = \bigcup_{B \in \mathcal B} B$ and for every $B_1, B_2 \in \mathcal B$ and for every $x \in B_1 \cap B_2$ there exists a $B \in \mathcal B$ such that:
(3)
\begin{align} \quad x \in B \subseteq B_1 \cap B_2 \end{align}
  • If the set $\mathcal B$ satisfies these two conditions, the the Topology Generated by the Basis $\mathcal B$, call it $\tau$, is the set of all unions of elements from $\mathcal B$:
(4)
\begin{align} \quad \tau = \left \{ U = \bigcup_{B \in \mathcal B^*} B : \mathcal B^* \subseteq \mathcal B \right \} \end{align}
  • Hence, if the collection $\mathcal B$ satisfies the two conditions stated above then it is a basis for the topology generated by $\mathcal B$ which can be entirely described by $\mathcal B$.
(5)
\begin{align} \quad \mathcal B = \{ [a, b) : a, b \in \mathbb{R}, a < b \} \end{align}
  • Meanwhile, the Upper Limit Topology is defined to be the topology generated by:
(6)
\begin{align} \quad \mathcal B = \{ (a, b] : a, b \in \mathbb{R}, a < b \} \end{align}
  • We verified that indeed these sets satisfy the two conditions for the collection of all unions from these sets (the generated topologies) to actually be topologies on $\mathbb{R}$.
  • We then took a quick break and on the Comparable Topologies on a Set we said that if $(X, \tau_1)$ and $(X, \tau_2)$ are both topological spaces then the topologies $\tau_1$ and $\tau_2$ are said to be Comparable if either $\tau_1 \subseteq \tau_2$ or $\tau_1 \supseteq \tau_2$. Furthermore, if $\tau_1 \subseteq \tau_2$ then $\tau_2$ is said to be Finer than $\tau_1$ and $\tau_1$ is said to be Coarser than $\tau_2$.
  • On the Local Bases of a Point in a Topological Space we said that a Local Basis of the point $x \in X$ is a collection of open neighbourhoods of $x$, $\mathcal B_x$ such for every open neighbourhood of $x$ there contains a $B \in \mathcal B_x$ contained in $U$, i.e., for all $U \in \tau$ with $x \in U$ there exists a $B \in \mathcal B_x$ such that:
(7)
\begin{align} \quad x \in B \subseteq U \end{align}
(8)
\begin{align} \quad \mathcal B_x = \{ B \in \mathcal B : x \in B \} \end{align}
  • The second important theorem that we looked at is that if $\{ B_x \}_{x \in X}$ is a collection of local bases for all $x \in X$, then we can construct a basis for the topology $\tau$ as the union of all of these local bases, i.e.,
(9)
\begin{align} \quad \mathcal B = \bigcup_{x \in X} \mathcal B_x \end{align}
  • After that, on the First Countable Topological Spaces we said that $(X, \tau)$ is a First Countable Topological Space if every $x \in X$ has a countable local basis. We showed that the set of real numbers with the usual topology is a first countable topological space and noted that every metric space is a first countable topological space. Additionally, if $X$ is a finite set, then $(X, \tau)$ must be a first countable topological space regardless of the topology $\tau$ on $X$.
  • On the Second Countable Topological Spaces page we then said that $(X, \tau)$ is a Second Countable Topological Space if there exists a countable basis for the topology $\tau$. Once again, we noted that if $X$ is finite then it must be a second countable topological space.
  • We then looked an amazingly simple and important theorem on the All Second Countable Topological Spaces are First Countable page, which (like the title implies) states that every second countable topological space is also a first countable topological space. We proved this by noting that if $\mathcal B$ is a countable basis for the topological space $(X, \tau)$ then for each $x \in X$ we can take $\mathcal B_x = \{ B \in \mathcal B : x \in B \}$ to be a local basis of $x$, but then $\mathcal B_x \subseteq \mathcal B$, so each of the local bases for the points $x \in X$ are countable.
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