Topology 2

There is an alternate definition of the basis generated by a topology that is often useful:

Theorem 1 Let {\mathcal B} be a basis for a topology {\mathcal T} on {X}. Then {O\in\mathcal T} if and only if

\displaystyle O=\bigcup_{\alpha\in I}U_\alpha

where {\{U_\alpha : \alpha\in I \}\subset\mathcal B}.

Proof: Let {O\in\mathcal T}. Then for each {x\in O}, there exists {U_x\in\mathcal T} such that {x\in U\subset O}. It follows that {O=\bigcup_{x\in X}U_x}. Conversely, if {O} is the union of open sets {U_\alpha}, then for each {x\in O}, there is a {\beta} such that {x\in U_\beta}, and so there is a basis element {B\in\mathcal B} with {x\in B\subset U_\beta\subset O}, which implies that {O\in\mathcal T}. \Box

Bases are convenient for defining topologies whose members cannot easily be generically expressed. For example, given a metric space {(X,d)}, the collection of open balls, that is, sets of the form

\displaystyle B(x,r)=\{y\in X: d(x,y)<r\}

is a basis for topology on {X}. For clearly

\displaystyle X=\bigcup_{x\in X}B(x,1),

and if {x\in B(y,r)\cap B(z,q)} then {B(x, r-d(x,y))\subset B(y,r)} and {B(x, q-d(x,z))\subset B(z,q)}, so letting {r^*=\min\{r-d(x,y), q-d(x,z)\}}, we have

\displaystyle x\in B(x, r^*)\subset B(y,r)\cap B(z,q).

Indeed, the “usual topology” on {\mathbb R}, with a set {U\subset\mathbb R} being open iff for each {x\in U} there exists {\varepsilon_x>0} such that {|x-y|<\varepsilon_x} implies {y\in U}, is generated by this basis. If {U} is open in {\mathbb R}, then

\displaystyle U=\bigcup_{x\in U} \left\{y\in \mathbb R:|x-y|<\varepsilon_x \right\},

that is, open sets in {\mathbb R} are unions of open balls.

More generally, there is the notion of a subbasis:

Definition 2 A collection of subsets {\mathcal S} of {X} is a subbasis for a topology on {X} if {X = \bigcup_{S\in\mathcal S}S}. The topology generated by {\mathcal S} is the collection of sets of the form

\displaystyle \bigcup_{T\subset S}\bigcap_{\substack{S\subset\mathcal S\\ S\text{finite}}} T.

In particular, given a family of topological spaces {\{(X_\alpha,\mathcal T_\alpha):\alpha\in I\}}, we define the product topology as that generated by the subbasis

\displaystyle \mathcal S = \bigcup_{\alpha\in I} \{\pi_\alpha^{-1}(U): U\in\mathcal T_\alpha\},

where

\displaystyle \pi_\beta:\bigcup_{\alpha\in I} X_\alpha \rightarrow X_\beta

is the projection map defined by {\pi_\beta((x_\alpha)_{\alpha\in I}) = x_\beta}.

Theorem 3 The box topology is the same as the product topology when {I} is finite and strictly finer than the product topology when {I} is infinite.

Proof: If {|I|=n}, then for any {x\in \prod_{i=1}^n X_i=:X} and {U=\prod_{i=1}^n U_i} open in {X} with {x\in U}, we have

\displaystyle U=\prod_{i=1}^n \pi_i^{-1}(U_i),

so {U} is an element of the basis for the product topology.

If {I} is infinite, then an element {B} of the basis {\mathcal B} for the product topology is of the form

\displaystyle B = \prod_{i=1}^n \pi_{\beta_i}^{-1}(U_{\beta_i}),

where {U_{\beta_i}} is open in {X_{\beta_i}}, so

\displaystyle B=\prod_{i\in I} V_\alpha,

where

\displaystyle V_{\alpha} = \begin{cases} \pi_{\alpha}^{-1}(U_\alpha),& \alpha\in\{\beta_1,\ldots,\beta_n\}\\ X_\alpha,& \text{ otherwise}. \end{cases}

If {(U_\alpha)_{\alpha\in I}} is open in {U_\alpha} for each {\alpha\in I}, then

\displaystyle U:=\prod_{\alpha\in i}U_\alpha

is an element of {\beta} if and only if {U_\alpha=X_\alpha} for all but finitely many {\alpha}.

\Box

For example, the box topology on {\mathbb R^n} is generated by the open rectangles

\displaystyle \prod_{i=1}^n (x_i-\varepsilon_i, x_i+\varepsilon_i).

If {U=\prod_{i=1}^n (x_i-\varepsilon_i, x_i+\varepsilon_i)} then

\displaystyle U=\prod_{i=1}^n \pi_i^{-1}((x_i-\varepsilon_i, x_i+\varepsilon_i)),

is also open in the product topology.

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