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| In [[mathematics]], a '''base''' (or '''basis''') ''B'' for a [[topological space]] ''X'' with [[topological space|topology]] ''T'' is a collection of [[open set]]s in ''T'' such that every open set in ''T'' can be written as a union of elements of ''B''.<ref>{{cite book |last1=Merrifield |first1=Richard E. |last2=Simmons |first2=Howard E. |authorlink2=Howard Ensign Simmons, Jr. |title=Topological Methods in Chemistry |year=1989 |publisher=John Wiley & Sons |location=New York |isbn=0-471-83817-9 |url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471838179.html |accessdate=27 July 2012 |pages=16 |quote='''Definition.''' A collection ''B'' of subsets of a topological space ''(X,T)'' is called a ''basis'' for ''T'' if every open set can be expressed as a union of members of ''B''.}}</ref><ref>{{cite book |last=Armstrong |first=M. A. |title=Basic Topology |year=1983 |publisher=Springer |isbn=0-387-90839-0 |url=https://www.springer.com/mathematics/geometry/book/978-0-387-90839-7 |accessdate=13 June 2013 |page=30 |quote=Suppose we have a topology on a set ''X'', and a collection <math>\beta</math> of open sets such that every open set is a union of members of <math>\beta</math>. Then <math>\beta</math> is called a ''base'' for the topology...}}</ref> We say that the base ''generates'' the topology ''T''. Bases are useful because many properties of topologies can be reduced to statements about a base generating that topology, and because many topologies are most easily defined in terms of a base which generates them.
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| == Simple properties of bases ==
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| Two important properties of bases are:
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| # The base elements ''[[cover (topology)|cover]]'' ''X''.
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| # Let ''B''<sub>1</sub>, ''B''<sub>2</sub> be base elements and let ''I'' be their intersection. Then for each ''x'' in ''I'', there is a base element ''B''<sub>3</sub> containing ''x'' and contained in ''I''.
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| If a collection ''B'' of subsets of ''X'' fails to satisfy either of these, then it is not a base for ''any'' topology on ''X''. (It is a [[subbase]], however, as is any collection of subsets of ''X''.) Conversely, if ''B'' satisfies both of the conditions 1 and 2, then there is a unique topology on ''X'' for which ''B'' is a base; it is called the topology '''generated''' by ''B''. (This topology is the [[intersection (set theory)|intersection]] of all topologies on ''X'' containing ''B''.) This is a very common way of defining topologies. A sufficient but not necessary condition for ''B'' to generate a topology on ''X'' is that ''B'' is closed under intersections; then we can always take ''B''<sub>3</sub> = ''I'' above.
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| For example, the collection of all [[open interval]]s in the [[real line]] forms a base for a topology on the real line because the intersection of any two open intervals is itself an open interval or empty.
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| In fact they are a base for the standard topology on the [[real number]]s.
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| However, a base is not unique. Many bases, even of different sizes, may generate the same topology. For example, the open intervals with rational endpoints are also a base for the standard real topology, as are the open intervals with irrational endpoints, but these two sets are completely disjoint and both properly contained in the base of all open intervals. In contrast to a [[basis (linear algebra)|basis]] of a [[vector space]] in [[linear algebra]], a base need not be [[maximal]]; indeed, the only maximal base is the topology itself. In fact, any open sets in the space generated by a base may be safely added to the base without changing the topology. The smallest possible [[cardinality]] of a base is called the '''weight''' of the topological space.
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| An example of a collection of open sets which is not a base is the set ''S'' of all semi-infinite intervals of the forms (−∞, ''a'') and (''a'', ∞), where ''a'' is a real number. Then ''S'' is ''not'' a base for any topology on '''R'''. To show this, suppose it were. Then, for example, (−∞, 1) and (0, ∞) would be in the topology generated by ''S'', being unions of a single base element, and so their intersection (0,1) would be as well. But (0, 1) clearly cannot be written as a union of the elements of ''S''. Using the alternate definition, the second property fails, since no base element can "fit" inside this intersection.
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| Given a base for a topology, in order to prove convergence of a net or sequence it is sufficient to prove that it is eventually in every set in the base which contains the putative limit.
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| == Objects defined in terms of bases ==
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| * The [[order topology]] is usually defined as the topology generated by a collection of open-interval-like sets.
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| * The [[metric topology]] is usually defined as the topology generated by a collection of [[open ball]]s.
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| * A [[second-countable space]] is one that has a [[countable]] base.
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| * The [[discrete topology]] has the [[Singleton (mathematics)|singletons]] as a base.
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| * The [[Residually finite group|profinite topology]] on a group is defined by taking the collection of all normal subgroups of finite index as a basis of open neighborhoods of the identity.
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| == Theorems ==
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| * For each point ''x'' in an open set ''U'', there is a base element containing ''x'' and contained in ''U''.
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| * A topology ''T''<sub>2</sub> is [[topological space|finer]] than a topology ''T''<sub>1</sub> [[if and only if]] for each ''x'' and each base element ''B'' of ''T''<sub>1</sub> containing ''x'', there is a base element of ''T''<sub>2</sub> containing ''x'' and contained in ''B''.
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| * If ''B''<sub>1</sub>,''B''<sub>2</sub>,...,''B''<sub>''n''</sub> are bases for the topologies ''T''<sub>1</sub>,''T''<sub>2</sub>,...,''T''<sub>''n''</sub>, then the [[Cartesian product|set product]] ''B''<sub>1</sub> × ''B''<sub>2</sub> × ... × ''B''<sub>''n''</sub> is a base for the [[product topology]] ''T''<sub>1</sub> × ''T''<sub>2</sub> × ... × ''T''<sub>''n''</sub>. In the case of an infinite product, this still applies, except that all but finitely many of the base elements must be the entire space.
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| * Let ''B'' be a base for ''X'' and let ''Y'' be a [[topological space|subspace]] of ''X''. Then if we intersect each element of ''B'' with ''Y'', the resulting collection of sets is a base for the subspace ''Y''.
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| * If a function ''f'':''X'' → ''Y'' maps every base element of ''X'' into an open set of ''Y'', it is an [[open map]]. Similarly, if every preimage of a base element of ''Y'' is open in ''X'', then ''f'' is [[continuity (topology)|continuous]].
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| * A collection of subsets of ''X'' is a topology on ''X'' if and only if it generates itself.
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| * ''B'' is a basis for a topological space ''X'' if and only if the subcollection of elements of ''B'' which contain ''x'' form a [[local base]] at ''x'', for any point ''x'' of ''X''.
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| ==Base for the closed sets==
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| [[Closed set]]s are equally adept at describing the topology of a space. There is, therefore, a dual notion of a base for the closed sets of a topological space. Given a topological space ''X'', a '''base for the closed sets''' of ''X'' is a family of closed sets ''F'' such that any closed set ''A'' is an [[intersection (set theory)|intersection]] of members of ''F''.
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| Equivalently, a family of closed sets forms a base for the closed sets if for each closed set ''A'' and each point ''x'' not in ''A'' there exists an element of ''F'' containing ''A'' but not containing ''x''.
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| It is easy to check that ''F'' is a base for the closed sets of ''X'' if and only if the family of [[complement (set theory)|complements]] of members of ''F'' is a base for the open sets of ''X''.
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| Let ''F'' be a base for the closed sets of ''X''. Then
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| #<big>∩</big>''F'' = ∅
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| #For each ''F''<sub>1</sub> and ''F''<sub>2</sub> in ''F'' the union ''F''<sub>1</sub> ∪ ''F''<sub>2</sub> is the intersection of some subfamily of ''F'' (i.e. for any ''x'' not in ''F''<sub>1</sub> or ''F''<sub>2</sub> there is an ''F''<sub>3</sub> in ''F'' containing ''F''<sub>1</sub> ∪ ''F''<sub>2</sub> and not containing ''x'').
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| Any collection of subsets of a set ''X'' satisfying these properties forms a base for the closed sets of a topology on ''X''. The closed sets of this topology are precisely the intersections of members of ''F''.
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| In some cases it is more convenient to use a base for the closed sets rather than the open ones. For example, a space is [[completely regular]] if and only if the [[zero set]]s form a base for the closed sets. Given any topological space ''X'', the zero sets form the base for the closed sets of some topology on ''X''. This topology will be finest completely regular topology on ''X'' coarser than the original one. In a similar vein, the [[Zariski topology]] on '''A'''<sup>''n''</sup> is defined by taking the zero sets of polynomial functions as a base for the closed sets.
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| == Weight and character == | |
| We shall work with notions established in {{harv|Engelking|1977|loc=pp. 12, 127--128}}. Fix <math>X\,</math> a topological space. We define the '''weight''' as <math>w(X)\triangleq\,</math> the minimum cardinality of a basis; we define the '''network weight''' as <math>nw(X)\triangleq\,</math> the minimum cardinality of a network; the '''character of a point''' <math>\chi(x,X)\triangleq\,</math> the minimum cardinality of a neighbourhood basis for <math>x\,</math> in <math>X\,</math>; and the '''character''' of <math>X\,</math> to be <math>\chi(X)\triangleq\sup\{\chi(x,X):x\in X\}\,</math>.
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| Here, a '''network''' is a family <math>\mathcal{N}\,</math> of sets, for which, for all points <math>x\,</math> and open neighbourhoods <math>U\ni x\,</math>, there is a <math>B\in\mathcal{N}\,</math> for which <math>x\in B\subseteq U\,</math>.
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| The point of computing the character and weight is useful to be able to tell what sort of bases and local bases can exist. We have following facts:
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| * <math>nw(X)\leq w(X)\,</math>.
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| * if <math>X\,</math> is discrete, then <math>w(X)=nw(X)=|X|\,</math>.
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| * if <math>X\,</math> is Hausdorff, then <math>nw(X)\,</math> is finite iff <math>X\,</math> is finite discrete.
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| * if <math>B\,</math> a basis of <math>X\,</math> then there is a basis <math>B'\subseteq B\,</math> of size <math>|B'|\leq w(X)\,</math>.
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| * if <math>N\,</math> a neighbourhood basis for <math>x\in X\,</math> then there is a neighbourhood basis <math>N'\subseteq N\,</math> of size <math>|N'|\leq \chi(x,X)\,</math>.
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| * if <math>f:X\to Y\,</math> is a continuous surjection, then <math>nw(Y)\leq w(X)\,</math>. (Simply consider the <math>Y\,</math>-network <math>f'''B\triangleq\{f''U:U\in B\}\,</math> for each basis <math>B\,</math> of <math>X\,</math>.)
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| * if <math>(X,\tau)\,</math> is Hausdorff, then there exists a weaker Hausdorff topology <math>(X,\tau')\,</math> so that <math>w(X,\tau')\leq nw(X,\tau)\,</math>. So ''a fortiori'', if <math>X\,</math> is also compact, then such topologies coincide and hence we have, combined with the first fact, <math>nw(X)=w(X)\,</math>.
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| * if <math>f:X\to Y\,</math> a continuous surjective map from a compact metrisable space to an Hausdorff space, then <math>Y\,</math> is compact metrisable.
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| The last fact comes from the fact that <math>f(X)\,</math> is compact Hausdorff, and hence <math>nw(f(X))=w(f(X))\leq w(X)\leq\aleph_{0}\,</math> (since compact metrisable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrisable exactly in case they are second countable. (An application of this, for instance, is that every path in an Hausdorff space is compact metrisable.)
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| === Increasing chains of open sets ===
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| Using the above notation, suppose that <math>w(X)\leq\kappa\,</math> some infinite cardinal.
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| Then there does not exist a strictly increasing sequence of open sets (equivalently
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| strictly decreasing sequence of closed sets) of length <math>\geq\kappa^{+}\,</math>.
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| To see this (without the axiom of choice), fix <math>(U_{\xi})_{\xi\in\kappa}\,</math> a basis of open sets. And suppose ''per contra'', that <math>(V_{\xi})_{\xi\in\kappa^{+}}\,</math> were a strictly increasing sequence of open sets. This means <math> (\forall{\alpha<\kappa^{+}})V_{\alpha}\setminus\bigcup_{\xi<\alpha}V_{\xi}\,</math> is non-empty. If <math>x\in V_{\alpha}\setminus\bigcup_{\xi<\alpha}V_{\xi}\,</math>, we may utilise the basis to find some <math>U_{\gamma}\,</math> with <math>x\in U_{\gamma}\subseteq V_{\alpha}\,</math>. In this way we may well-define a map, <math>f:\kappa^{+}\to\kappa\,</math> mapping each <math>\alpha\,</math> to the least <math>\gamma\,</math> for which <math>U_{\gamma}\subseteq V_{\alpha}\,</math>
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| and meets <math>V_{\alpha}\setminus\bigcup_{\xi<\alpha}V_{\xi}\,</math>. This map can be seen to be injective. (For otherwise there would be <math>\alpha<\beta\,</math> with <math>f(\alpha)=f(\beta)=\gamma\,</math>, say, which would further imply <math>U_{\gamma}\subseteq V_{\alpha}\,</math> but also meets <math>V_{\beta}\setminus\bigcup_{\xi<\alpha}\subseteq V_{\beta}\setminus V_{\alpha}\,</math> which is a contradiction.) But this would go to show that <math>\kappa^{+}\leq\kappa\,</math>, a contradiction.
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| == See also ==
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| * [[Subbase]]
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| * [[Gluing axiom]]
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| ==References==
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| {{Reflist}}
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| {{refbegin}}
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| * {{cite book | author=[[Ryszard Engelking|Engelking, Ryszard]] | title=General Topology | publisher=PWN, Warsaw | year=1977}}
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| * [[James Munkres]] (1975) ''Topology: a First Course''. Prentice-Hall.
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| * Willard, Stephen (1970) ''General Topology''. Addison-Wesley. Reprinted 2004, Dover Publications.
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| {{refend}}
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| {{DEFAULTSORT:Base (Topology)}}
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| [[Category:General topology]]
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