Editing Subbase (section)

==Examples==

The topology generated by any subset <math>\mathcal{S} \subseteq \{\varnothing, X\}</math> (including by the empty set <math>\mathcal{S} := \varnothing</math>) is equal to the trivial topology <math>\{\varnothing, X\}.</math>

If <math>\tau</math> is a topology on <math>X</math> and <math>\mathcal{B}</math> is a basis for <math>\tau</math> then the topology generated by <math>\mathcal{B}</math> is <math>\tau.</math> Thus any basis <math>\mathcal{B}</math> for a topology <math>\tau</math> is also a subbasis for <math>\tau.</math> 
If <math>\mathcal{S}</math> is any subset of <math>\tau</math> then the topology generated by <math>\mathcal{S}</math> will be a subset of <math>\tau.</math>

The usual topology on the [[real number]]s <math>\R</math> has a subbase consisting of all [[semi-infinite]] open intervals either of the form <math>(-\infty, a)</math> or <math>(b, \infty),</math> where <math>a</math> and <math>b</math> are real numbers. Together, these generate the usual topology, since the intersections <math>(a,b) = (-\infty, b) \cap (a, \infty)</math> for <math>a \leq b</math> generate the usual topology. A second subbase is formed by taking the subfamily where <math>a</math> and <math>b</math> are [[Rational number|rational]]. The second subbase generates the usual topology as well, since the open intervals <math>(a, b)</math> with <math>a,</math> <math>b</math> rational, are a basis for the usual Euclidean topology.

The subbase consisting of all semi-infinite open intervals of the form <math>(-\infty, a)</math> alone, where <math>a</math> is a real number, does not generate the usual topology. The resulting topology does not satisfy the [[T1 space|T<sub>1</sub> separation axiom]], since if <math>a < b</math> every [[open set]] containing <math>b</math> also contains <math>a.</math>

The [[initial topology]] on <math>X</math> defined by a family of functions <math>f_i : X \to Y_i,</math> where each <math>Y_i</math> has a topology, is the coarsest topology on <math>X</math> such that each <math>f_i</math> is [[Continuous function (topology)|continuous]]. Because continuity can be defined in terms of the [[Inverse image|inverse images]] of open sets, this means that the initial topology on <math>X</math> is given by taking all <math>f_i^{-1}(U),</math>
where <math>U</math> ranges over all open subsets of <math>Y_i,</math> as a subbasis.

Two important special cases of the initial topology are the [[product topology]], where the family of functions is the set of projections from the product to each factor, and the [[subspace topology]], where the family consists of just one function, the [[inclusion map]].

The [[compact-open topology]] on the space of continuous functions from <math>X</math> to <math>Y</math> has for a subbase the set of functions
<math display=block>V(K,U) = \{f : X \to Y \mid f(K) \subseteq U\}</math>
where <math>K \subseteq X</math> is [[Compact space|compact]] and <math>U</math> is an open subset of <math>Y.</math>

Suppose that <math>(X, \tau)</math> is a [[Hausdorff space|Hausdorff]] topological space with <math>X</math> containing two or more elements (for example, <math>X = \R</math> with the [[Euclidean topology]]). Let <math>Y \in \tau</math> be any non-empty {{em|open}} subset of <math>(X, \tau)</math> (for example, <math>Y</math> could be a non-empty bounded open interval in <math>\R</math>) and let <math>\nu</math> denote the [[subspace topology]] on <math>Y</math> that <math>Y</math> inherits from <math>(X, \tau)</math> (so <math>\nu \subseteq \tau</math>). Then the topology generated by <math>\nu</math> '''on <math>X</math>''' is equal to the union <math>\{X\} \cup \nu</math> (see the footnote for an explanation),
<ref group=note>Since <math>\nu</math> is a topology on <math>Y</math> and <math>Y</math> is an open subset of <math>(X, \tau),</math>, it is easy to verify that <math>\{X\} \cup \nu</math> is a topology on <math>X</math>. In particular, <math>\nu</math> is closed under unions and finite intersections because <math>\tau</math> is. But since <math> X \not\in \nu </math>, <math>\nu</math> is not a topology on <math>X</math> an <math>\{X\} \cup \nu</math> is clearly the smallest topology on <math>X</math> containing <math>\nu</math>).</ref>
where <math>\{X\} \cup \nu \subseteq \tau</math> (since <math>(X, \tau)</math> is Hausdorff, equality will hold if and only if <math>Y = X</math>). Note that if <math>Y</math> is a [[proper subset]] of <math>X,</math> then <math>\{X\} \cup \nu</math> is the smallest topology ''on <math>X</math>'' containing <math>\nu</math> yet <math>\nu</math> does not cover <math>X</math> (that is, the union <math>\bigcup_{V \in \nu} V = Y</math> is a proper subset of <math>X</math>).