Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Specializations:: *[[Specializations]]* Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* Examples:: [[unioned preimages of open sets under projections form subbasis for product topology]] ---- - Let $X$ be a set. > [!definition] Definition. ([[subbasis for a topology]]) > A **subbasis** $\mathscr{S}$ for a [[topological space|topology on]] $X$ is a collection of subsets of $X$ whose union equals $X$. The **topology generated by the subbasis $\mathscr{S}$** is defined to be the collection $\tau$ of all unions of finite intersections of elements of $\mathscr{S}$. > [!justification] We must check that $\tau$ is a [[topological space|topology on]] $X$. For this purpose it will suffice to show that the collection $\mathscr{B}$ of all *finite intersections* of elements of $\mathscr{S}$ is a [[basis for a topology|basis]]... for then when we consider the set $\tau$ of all unions of said finite intersections we know it is a [[topological space|topology]] on $X$ by [[open sets are unions of basis elements]] (Munkres 13.1). \ For $x \in X$, it is clear that there exists $U_{x}$ s.t. $x \in U_{x} \in \mathscr{S}$ by definition of $\mathscr{S}$. Now suppose that $x \in B_{1} \cap B_{2}$ for $B_{1}, B_{2} \in \mathscr{B}$, where $B_{1} = S_{1} \cap \dots \cap S_{n} \and B_{2}=S_{1}' \cap \dots \cap S_{n}'.$ Their intersection $B :=S_{1} \cap \dots \cap S_{n} \cap S_{1}' \cap \dots \cap S_{n} ' \in \mathscr{B}$ is also a finite intersection of elements in $\mathscr{S}$, and hence belongs to $\mathscr{B}$; that is, we have $x \in B \subset B_{1} \cap B_{2}$ and so the second basis condition is satisfied. ^c11575 > [!intuition] > The idea is that a subbasis 'generates a basis' (see the justification above), and a basis in turn [[topology generated by a basis|generates a topology]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```