Basis and Subbasis

Specifying every open set of a topology directly is often impractical. Instead, we generate topologies from smaller, more manageable collections of sets called bases and subbases. These provide efficient descriptions of topological spaces and are indispensable in constructing new topologies.

Basis for a Topology

Definition1.11Basis

Let $(X, \tau)$ be a topological space. A collection $\mathcal{B} \subseteq \tau$ is a basis for the topology $\tau$ if every open set $U \in \tau$ can be written as a union of elements of $\mathcal{B}$ : $U = \bigcup_{\alpha \in A} B_\alpha, \quad B_\alpha \in \mathcal{B}.$

Equivalently, $\mathcal{B}$ is a basis for $\tau$ if for every $U \in \tau$ and every $x \in U$ , there exists $B \in \mathcal{B}$ such that $x \in B \subseteq U$ .

Definition1.12Basis for a Topology (Intrinsic)

A collection $\mathcal{B}$ of subsets of a set $X$ is a basis for a topology on $X$ if:

Covering: For every $x \in X$ , there exists $B \in \mathcal{B}$ with $x \in B$ .
Refinement: For every $B_1, B_2 \in \mathcal{B}$ and every $x \in B_1 \cap B_2$ , there exists $B_3 \in \mathcal{B}$ such that $x \in B_3 \subseteq B_1 \cap B_2$ .

The topology $\tau_{\mathcal{B}}$ generated by such a basis $\mathcal{B}$ consists of all unions of elements of $\mathcal{B}$ : $\tau_{\mathcal{B}} = \left\{\bigcup_{\alpha \in A} B_\alpha : B_\alpha \in \mathcal{B}\right\}.$

ExampleStandard Bases

The collection of open intervals $\{(a, b) : a < b, \; a, b \in \mathbb{R}\}$ is a basis for the standard topology on $\mathbb{R}$ .
The collection of open intervals with rational endpoints $\{(p, q) : p < q, \; p, q \in \mathbb{Q}\}$ is also a basis for the standard topology on $\mathbb{R}$ . This shows $\mathbb{R}$ is second-countable.
The collection of open balls $\{B(x, r) : x \in \mathbb{R}^n, r > 0\}$ is a basis for the standard topology on $\mathbb{R}^n$ .
The collection of all singletons $\{\{x\} : x \in X\}$ is a basis for the discrete topology on $X$ .

Comparing Bases

RemarkTwo Bases and the Same Topology

Two bases $\mathcal{B}_1$ and $\mathcal{B}_2$ on $X$ generate the same topology if and only if:

For every $B_1 \in \mathcal{B}_1$ and $x \in B_1$ , there exists $B_2 \in \mathcal{B}_2$ with $x \in B_2 \subseteq B_1$ , and
For every $B_2 \in \mathcal{B}_2$ and $x \in B_2$ , there exists $B_1 \in \mathcal{B}_1$ with $x \in B_1 \subseteq B_2$ .

In other words, each basis element of one can be "refined" by basis elements of the other at every point.

ExampleLower Limit Topology (Sorgenfrey Line)

The collection $\mathcal{B} = \{[a, b) : a < b\}$ of half-open intervals is a basis for a topology on $\mathbb{R}$ called the lower limit topology (or Sorgenfrey line), denoted $\mathbb{R}_\ell$ .

This topology is strictly finer than the standard topology: every open interval $(a, b) = \bigcup_{n} [a + 1/n, b)$ is open in $\mathbb{R}_\ell$ , but $[0, 1)$ is open in $\mathbb{R}_\ell$ and not in the standard topology.

The Sorgenfrey line is an important source of counterexamples:

$\mathbb{R}_\ell$ is separable (rationals are dense) and first-countable.
$\mathbb{R}_\ell$ is not second-countable.
$\mathbb{R}_\ell \times \mathbb{R}_\ell$ (the Sorgenfrey plane) is not normal, despite $\mathbb{R}_\ell$ being normal.

Subbasis for a Topology

Definition1.13Subbasis

A collection $\mathcal{S}$ of subsets of $X$ is a subbasis for a topology on $X$ if $\mathcal{S}$ covers $X$ , i.e., $\bigcup_{S \in \mathcal{S}} S = X$ .

The topology $\tau_{\mathcal{S}}$ generated by $\mathcal{S}$ is defined as the coarsest topology on $X$ containing $\mathcal{S}$ . Explicitly: $\tau_{\mathcal{S}} = \left\{ \bigcup_{\alpha \in A} \left( \bigcap_{i=1}^{n_\alpha} S_{\alpha,i} \right) : S_{\alpha,i} \in \mathcal{S}, \; n_\alpha \in \mathbb{Z}^+ \right\}.$

In other words, the topology generated by $\mathcal{S}$ has as a basis all finite intersections of elements of $\mathcal{S}$ .

ExampleSubbasis Examples

The collection $\mathcal{S} = \{(-\infty, b) : b \in \mathbb{R}\} \cup \{(a, \infty) : a \in \mathbb{R}\}$ is a subbasis for the standard topology on $\mathbb{R}$ . The finite intersections of subbasis elements yield all open intervals: $(a, b) = (a, \infty) \cap (-\infty, b)$ .
In the product topology on $\prod_{\alpha} X_\alpha$ , the canonical subbasis consists of all sets $\pi_\alpha^{-1}(U_\alpha)$ where $U_\alpha$ is open in $X_\alpha$ .

RemarkSubbasis vs. Basis

Every basis is a subbasis, but not conversely. The subbasis approach is the most economical way to specify a topology. It is particularly important in:

Product topologies: defined via the subbasis of preimages under projections.
Weak topologies: the coarsest topology making a given family of maps continuous.
Alexander's subbasis theorem: compactness can be checked using subbasis covers only (see Chapter 8).

The Order Topology

Definition1.14Order Topology

Let $(X, \leq)$ be a totally ordered set. The order topology on $X$ is the topology generated by the subbasis consisting of all open rays: $\mathcal{S} = \{(a, \infty) : a \in X\} \cup \{(-\infty, b) : b \in X\}$ where $(a, \infty) = \{x \in X : x > a\}$ and $(-\infty, b) = \{x \in X : x < b\}$ .

If $X$ has a minimum element $a_0$ , include $[a_0, b)$ for all $b > a_0$ . If $X$ has a maximum element $b_0$ , include $(a, b_0]$ for all $a < b_0$ .

ExampleOrder Topology Examples

The order topology on $\mathbb{R}$ coincides with the standard topology.
The order topology on $\{0, 1\} \times \mathbb{Z}^+$ in dictionary order gives the discrete topology.
The long line $L$ is constructed by taking $\omega_1 \times [0,1)$ in the dictionary order topology (where $\omega_1$ is the first uncountable ordinal), then removing the minimum element. It is connected, locally homeomorphic to $\mathbb{R}$ , but not metrizable.