# Sub-base

A sub-base is in the basic mathematical discipline of set topology , a special class of sets of open sets . A sub-base uniquely defines a topology and thus often simplifies proofs, since it is sufficient to limit oneself to the sets of the sub-base. Likewise, some properties of topologies are also defined as properties of their sub-bases.

Conversely, every set system can be understood as a sub-basis and thus enables targeted topologies with specific properties to be constructed.

In the literature translated from Russian into English there is also the term "Pre-Base" (German: Pre-Basis ) instead of the typical English terms subbase or subbasis .

## definition

The conventions apply

${\ displaystyle \ bigcap _ {j \ in \ emptyset} A_ {j} = X}$and .${\ displaystyle \ bigcup _ {j \ in \ emptyset} A_ {j} = \ emptyset}$

A topological space and a set system are given . Then a sub-base of the topology is called if one of the following equivalent conditions is met: ${\ displaystyle (X, {\ mathcal {O}})}$ ${\ displaystyle {\ mathcal {S}} \ subset {\ mathcal {O}}}$${\ displaystyle {\ mathcal {S}}}$${\ displaystyle {\ mathcal {O}}}$

• Every open set is the union of any number of sets that are themselves intersections of finitely many sets .${\ displaystyle O \ in {\ mathcal {O}}}$${\ displaystyle {\ mathcal {S}}}$
• The set of all cuts from finitely many sets , so${\ displaystyle {\ mathcal {S}}}$
${\ displaystyle {\ mathcal {B}}: = \ lbrace M \ subset X \, | \, M = \ bigcap _ {j \ in J} S_ {j}, \; S_ {j} \ in {\ mathcal {S}}, \; | J | <\ infty \ rbrace}$
forms a basis of the topology .${\ displaystyle {\ mathcal {O}}}$
• ${\ displaystyle {\ mathcal {S}}}$generated in the sense that${\ displaystyle {\ mathcal {O}}}$
• ${\ displaystyle {\ mathcal {O}}}$is the smallest topology (with respect to the subset relationship) that contains and${\ displaystyle {\ mathcal {S}}}$
• any further topology that contains is always finer than .${\ displaystyle {\ mathcal {S}}}$${\ displaystyle {\ mathcal {O}}}$

## Examples

If there is an infinite set, then the set forms all finite subsets of a given, finite power , i.e. ${\ displaystyle X}$${\ displaystyle n \ neq 0}$

${\ displaystyle {\ mathcal {S}}: = \ {M \ subset X \, | \, | M | = n \}}$

a sub-base of the discrete topology given by . Because it applies to selecting suitable from that given for one . Thus, all single-element subsets of can be generated from. These then form the basis of the discrete topology. ${\ displaystyle {\ mathcal {O}} _ {D}: = {\ mathcal {P}} (X)}$${\ displaystyle S_ {1}, S_ {2}}$${\ displaystyle {\ mathcal {S}}}$${\ displaystyle S_ {1} \ cap S_ {2} = \ {x \}}$${\ displaystyle x \ in X}$${\ displaystyle {\ mathcal {S}}}$${\ displaystyle X}$

A sub-basis of the natural topology on the real numbers is given by

${\ displaystyle {\ mathcal {S}}: = {\ mathcal {S}} ^ {+} \ cup {\ mathcal {S}} ^ {-}}$,

in which

${\ displaystyle {\ mathcal {S}} ^ {-}: = \ {(- \ infty, b) \, | \, b \ in \ mathbb {R} \}}$ and ${\ displaystyle {\ mathcal {S}} ^ {+}: = \ {(a, + \ infty) \, | \, a \ in \ mathbb {R} \}}$

is. This is because the set of open intervals forms a basis of the natural topology, and every open interval can be passed through from the sub-basis

${\ displaystyle (a, b) = (- \ infty, b) \ cap (a, + \ infty)}$

produce.

## properties

### Ambiguity

Although sub-bases clearly determine the topology, a topology generally has more than one sub-base. So form both

${\ displaystyle {\ mathcal {S}} _ {1}: = \ {\ {1 \}, \ {2 \}, \ {3 \} \}}$ as well as
${\ displaystyle {\ mathcal {S}} _ {2}: = \ {\ {1,2 \}, \ {2,3 \}, \ {3,1 \} \}}$

a sub base of . Likewise, the natural topology does not merely have the sub-base given above as an example. For example, it is also sufficient to consider intervals of the form and for rational interval limits, i.e. for . ${\ displaystyle {\ mathcal {O}} = {\ mathcal {P}} (\ {1,2,3 \})}$${\ displaystyle \ mathbb {R}}$${\ displaystyle (- \ infty, a)}$${\ displaystyle (b, + \ infty)}$${\ displaystyle a, b \ in \ mathbb {Q}}$

### Generation of topologies through sub-bases

Just as a topology determines its sub-bases, a topology can also be determined by a sub-base. To do this, one chooses any set system and declares this to be the sub-base of a topology that is not yet more precisely specified. It should be noted here that, in contrast to the analogous procedure with bases, this is possible without any requirement for the quantity system. ${\ displaystyle {\ mathcal {M}}}$

This procedure, which is reflected in the third of the definitions given above, is formalized by the envelope operator

${\ displaystyle \ tau ({\ mathcal {M}}): = \ bigcap \ {{\ mathcal {E}} \ subseteq {\ mathcal {P}} (X) \, | \, {\ mathcal {M} } \ subseteq {\ mathcal {E}}, \, {\ mathcal {E}} {\ text {is topology on}} X \}}$.

This shell operator again provides a topology, since the intersection of topologies is again a topology. Furthermore, this topology is the coarsest topology that contains the specified system of quantities . ${\ displaystyle {\ mathcal {M}}}$

## Important statements using sub-bases

• The initial topology of a family of mappings from into the topological spaces is precisely the topology on whose sub-base consists of the archetypes of open sets, i.e. of for . Since both the subspace topology and the product topology are special cases of the initial topology, these topologies can also be defined via their sub-bases.${\ displaystyle (f_ {i}) _ {i \ in I}}$${\ displaystyle X}$${\ displaystyle (Y_ {i}, {\ mathcal {O}} _ {i})}$${\ displaystyle X}$${\ displaystyle f_ {i} ^ {- 1} (O_ {i})}$${\ displaystyle O_ {i} \ in {\ mathcal {O}} _ {i}}$
• Alexander theorem : It suffices to check compactness for sets from a sub-basis.
• It is also sufficient to check continuity on a sub-basis. So if is a mapping from to and an arbitrary sub-basis of , then is continuous if and only if is.${\ displaystyle f}$${\ displaystyle (X_ {1}, {\ mathcal {O}} _ {1})}$${\ displaystyle (X_ {2}, {\ mathcal {O}} _ {2})}$${\ displaystyle {\ mathcal {S}}}$${\ displaystyle {\ mathcal {O}} _ {2}}$${\ displaystyle f}$${\ displaystyle f ^ {- 1} ({\ mathcal {S}}) \ subset {\ mathcal {O}} _ {1}}$