Section 2 Basis, Density, and Size
Definition 2.1.
Let \(\tuple{X,\mc T}\) be a topological space. A subset \(\mc B\subseteq\mc T\) is called a basis for the topology if for every \(x\in X\) and neighborhood \(U\) of \(x\text{,}\) there exists \(B\in\mc B\) such that \(x\in B\subseteq U\text{.}\)
Proposition 2.2.
\(\mc B=\setBuilder{(a,b)}{a,b\in\mb R}\) is a basis for the Euclidean topology.
Theorem 2.3.
Let \(\mc B\subseteq\mc P(X)\) satisfy the following properties:
For all \(x\in X\text{,}\) there exists \(B\in\mc B\) such that \(x\in B\text{.}\)
If \(x\in A\in\mc B\) and \(x\in B\in\mc B\text{,}\) there exists \(C\in\mc B\) such that \(x\in C\subseteq A\cap B\text{.}\)
Then \(\mc T=\setBuilder{\bigcup\mc U}{\mc U\subseteq\mc B}\) is a topology, and \(\mc B\) is a basis for that topology. We call this the topology generated by the basis.
Proposition 2.4.
\(\mc B=\setBuilder{[a,b)}{a,b\in\mb R}\) is a basis for a topology different from the Euclidean topology, called the Sorgenfrey topology.
Checkpoint 2.5. Examples of bases.
Calculate the topology generated by each basis on \(\mb R\text{.}\)
\(\displaystyle \mc B=\setBuilder{(a,b)}{a,b\in\mb Q}\)
\(\displaystyle \mc B=\setBuilder{(a,\infty)}{a\in\mb R}\)
\(\displaystyle \mc B=\setBuilder{\setList{x}}{x\in\mb R}\)
\(\displaystyle \mc B=\setBuilder{[a,b]}{a,b\in\mb R}\)
\(\displaystyle \mc B=\setBuilder{[a,b]}{a,b\in\mb R,a\lt0\lt b}\)
Theorem 2.6.
Two bases \(\mc B_0,\mc B_1\) generate the same topology if and only if for all \(x\in B_0\in\mc B_0\) there exists \(B_1\in\mc B_1\) such that \(x\in B_1\subseteq B_0\text{,}\) and for all \(x\in B_1\in\mc B_1\) there exists \(B_0\in\mc B_0\) such that \(x\in B_0\subseteq B_1\text{.}\)
Theorem 2.7.
Let \(\mc S\subseteq\mc P(X)\) and
\begin{equation*}
\mc T=\bigcap\setBuilder{\mc T^\star\subseteq\mc P(X)}{\mc S\subseteq\mc T^\star \text{ and }
\mc T^\star \text{ is a topology on } X}.
\end{equation*}
Then \(\mc T\) is a topology.
Definition 2.8.
The set
\(\mc S\subseteq\mc P(X)\) in
Theorem 2.7 is called a
subbasis generating the topology
\begin{equation*}
\mc T=\bigcap\setBuilder{\mc T^\star\subseteq\mc P(X)}{\mc S\subseteq\mc T^\star \text{ and }
\mc T^\star \text{ is a topology on } X}.
\end{equation*}
Checkpoint 2.9. Topologies generated from subbases.
Calculate the topology on \(\mb R\) generated by each subbasis.
\(\displaystyle \setBuilder{(-\infty,x)}{x\in\mb R}\cup\setBuilder{(y,\infty)}{y\in\mb R}\)
\(\displaystyle \setBuilder{(-\infty,x]}{x\in\mb R}\cup\setBuilder{[y,\infty)}{y\in\mb R}\)
\(\displaystyle \setList{\setList{0}}\)
\(\displaystyle \mc T\cup\setList{\mb R\setminus\setBuilder{\frac{1}{2^n}}{n\in\omega}}\)
Theorem 2.10.
Let \(\mc S\subseteq\mc P(X)\) and
\begin{equation*}
\mc B=\setList{X}\cup\bigcap\setBuilder{\mc B^\star\subseteq\mc P(X)}
{\mc S\subseteq\mc B^\star\text{ and }
B_1,B_2\in\mc B^\star\Rightarrow B_1\cap B_2\in\mc B^\star}.
\end{equation*}
Then \(\mc B\) is a basis for a topology on \(X\text{,}\) and the topology generated by the basis \(\mc B\) is same as the topology generated by the subbasis \(\mc S\text{.}\)
Definition 2.11.
A subset \(D\subseteq X\) of a topological space is called dense if and only if \(\cl D=X\text{.}\)
Checkpoint 2.12.
Determine which of these are dense subsets of \(\mb R\text{.}\)
\(\displaystyle \mb Q\)
\(\displaystyle \mb Z\)
\(\displaystyle \mb R\setminus\mb Q\)
\(\displaystyle \mb R\setminus\mb Z\)
Theorem 2.13.
A subset \(D\) of a topological space is dense if and only if every nonempty open set of the space contains a point of \(D\text{.}\)
Proposition 2.14.
Let \(X\) be a topological space, and let \(D\subseteq E\subseteq X\text{.}\) If \(D\) is dense, then \(E\) is also dense.
Definition 2.15.
A space is said to have countable weight (or be second-countable) if it has a basis whose cardinality is countable.
Definition 2.16.
A space is said to have
countable density (or be
separable, no relation to
Section 4) if it has a dense subset whose cardinality is countable.
Theorem 2.17.
There exists a space with countable cardinality, but uncountable weight.
Example 2.18.
The reals with the Euclidean topology are a space with uncountable cardinality, but countable weight and countable density.
Theorem 2.19.
Every space with countable weight has countable density.
Example 2.20.
The reals with the Sorgenfrey topology are a space with uncountable cardinality and weight, but countable density.
Proposition 2.21.
The reals with the discrete topology are a space with uncountable cardinality, weight, and density.
Checkpoint 2.22.
Find or define a non-discrete topological space which has uncountable weight and density.
Proposition 2.23.
Let \(\mathcal B\) be a basis for \(X\) and \(Y\subseteq X\text{.}\) Then \(\mathcal B_Y=\{U\cap Y:U\in\mathcal B\}\) is a basis for the subspace topology on \(Y\text{.}\)
