J. Munkres. Topology (2013) Chapter 2: Topological Spaces and Continuous Functions. 13: Basis for a Topology

Definition

Let \(X\) be a set. A basis for a topology on \(X\) is a set \(\mathcal{B} \subseteq \mathcal{P} X\) such that:

  1. \( \bigcup \mathcal{B} = X\)
  2. For all \(B_1, B_2 \in \mathcal{B}\) and \(x \in B_1 \cap B_2\), there exists \(B_3 \in \mathcal{B}\) such that \(x \in B_3 \subseteq B_1 \cap B_2\).

The topology generated by \(\mathcal{B}\) is then \( \{U \in \mathcal{P} X \mid \forall x \in U. \exists B \in \mathcal{B}. x \in B \subseteq U \}\). This is the same as \( \{ \bigcup \mathcal{B}_0 \mid \mathcal{B}_0 \subseteq \mathcal{B} \}\).

Lemma 13.2

Let \(X\) be a topological space. Let \(\mathcal{B}\) be a set of open sets of \(X\) such that, for every open set \(U\) and each \(x \in U\), there exists \(B \in \mathcal{B}\) such that \(x \in B \subseteq U\). Then \(\mathcal{B}\) is a basis for the topology on \(X\).

Lemma 13.3

Let \(\mathcal{B}\) and \(\mathcal{B}'\) be bases for the topologies \(\mathcal{T}\) and \(\mathcal{T}'\) respectively. Then \(\mathcal{T} \subseteq \mathcal{T}'\) if and only if, for all \(B \in \mathcal{B}\) and \(x \in B\), there exists \(B' \in \mathcal{B}'\) such that \(x \in B' \subseteq B\).

Definition

The standard topology on \(\mathbb{R}\) is the topology generated by the set of all open intervals.

The lower limit topology on \(\mathbb{R}\) is the topology generated by the set of all half-open intervals of the form \([a,b)\). We write \(\mathbb{R}_l\) for the topological space consisting of \(\mathbb{R}\) under this topology.

Let \(K = \{ 1/n \mid n \in \mathbb{Z}_+ \}\). The \(K\)-topology on \(\mathbb{R}\) is the topology generated by the set of all open intervals \((a,b)\) and all sets of the form \((a,b) - K\) for \(a,b \in \mathbb{R}\), \(a < b\). We write \(\mathbb{R}_K\) for \(\mathbb{R}\) under this topology.

Lemma 13.4

The lower-limit topology and \(K\)-topology are strictly finer than the standard topology, but incomparable with one another.

Definition

A subbasis for a topology on a set \(X\) is a set \(\mathcal{S} \subseteq \mathcal{P} X\) such that \(\bigcup \mathcal{S} = X\). The topology generated by \(\mathcal{S}\) is then the set of all unions of finite intersections of elements of \(\mathcal{S}\).

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