Difference between basis and sub basis of a topological space.

Question

In munkrres there is lemma which says that "each open set in a topology is Union of its basis elements"or in other words we can say that a topology can be generated by Union of basis elements while in case of sub basis it is a subset of basis such that finite intersection and Union of these sets generate the topology....Now my question is that we many times even use finite intersection of basis elements also to generate topology as for example in case of discrete Topology all singletons constitutes a basis for this topology,now if we want to generate ∅ it can always be generated only by taking intersection of any two basis elements.so we used intersection here too,then what's the difference between basis and sub basis? Hope u understand my question Thanks in advance

Answer 1

"it can always be generated only by taking intersection of any two basis elements..."

That is not true.

If $\mathcal B$ denotes a basis of a topological space then every open set can written as: $$\cup\mathcal V:=\bigcup_{V\in\mathcal V}V$$ for some subcollection $\mathcal V\subseteq\mathcal B$, so also the empty set.

If we take $\mathcal V=\varnothing$ then we get: $$\cup\mathcal V=\cup\varnothing=\varnothing$$

So we do not need any intersection for that.

In words: $\varnothing$ can be looked at as empty union of sets in $\mathcal B$.

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If $\tau$ denotes a topology and $\mathcal B$ is a basis of $\tau$ then:$$\tau=\{\cup\mathcal V\mid\mathcal V\subseteq\mathcal B\}$$and (as shown above) then $\varnothing=\cup\varnothing\in\tau$.

This is not true in general if $\mathcal B$ is a subbasis of $\tau$.

Answer 2

Indeed, a base $\mathcal{B}$ of a topology $\mathcal{T}$ is a collection of open subsets of $X$, such that every $O \in \mathcal{T}$ can be written as the union $O=\bigcup \mathcal{B}'$ where $\mathcal{B}' \subseteq \mathcal{B}$.

A subbase $\mathcal{S}$ for that topology is also a collection of open subsets of $X$ such that the set of all *finite intersections" of subfamilies of $\mathcal{S}$ forms a base $\mathcal{B}_{\mathcal{S}}$ for $\mathcal{T}$, as defined above. This includes all one-element intersections, so $O \in \mathcal{S}$ can be written as $\bigcap \{O\}$, so $\mathcal{S} \subseteq \mathcal{B}_{\mathcal{S}}$. There is no a priori limit on the number of subbase elements we intersect, except that it is a finite intersection only (otherwise we could have non-open sets in the generated base).

I personally (and many texts agree with me) also include $X=\bigcap \emptyset$ (where $\emptyset$ is seen as a (very) finite subfamily of $\mathcal{S}$) in the generated base $\mathcal{B}_{\mathcal{S}}$, but Munkres does not do that and demands from the outset that the union of $\mathcal{S}$ is $X$ already, to avoid the empty intersection explanation (which might be a didactic choice one can defend).

For any base $\mathcal{B}$ we can always write $\emptyset = \bigcup \emptyset$ where $\emptyset \subseteq \mathcal{B}$ so the empty set we get "for free" in the topoloyg generated by a base.