1st December 2004

Munkres §19

Ex. 19.7. Any nonempty basis open set in the product topology contains an element from R^{∞}, cf. Example 7p. 151. Therefore \overline{R^{∞}} = R^{ω} in the product topology. (R^{∞} is dense [Definition p. 191] in R^{ω} with the product topology.)

Let (x_{i}) be any point in R^{ω} − R^{∞}. Put

U_{i} = \begin{cases}R & if x_{i} = 0 \\ R − \{0\} & if x_{i} ≠ 0 \end{cases}

Then ∏ U_{i} is open in the box topology and (x_{i}) ∈ ∏ U_{i} ⊂ R^{ω} − R^{∞}. This shows that R^{∞} is closed so that \overline{R^{∞}} = R^{∞} with the box topology on R^{ω}.

See [Ex 20.5] for the closure of R^{∞} in R^{ω} with the uniform topology.

Ex. 19.10.

(a). The topology T (the initial topology for the set maps {f_{α} | α ∈ J}) is the intersection [Ex 13.4] of all topologies on A for which all the maps f_{α}, α ∈ J, are continuous.

(b). Since all the functions f_{α}: A → X_{α}, α ∈ J, are continuous, S = ⋃ S_{α} ⊂ T . The topology T_{S} generated by S, which is the coarsest topology containing S [Ex 13.5], is therefore also contained in T . On the other hand, T ⊂ T_{S}, for all the functions f_{α}: A → X_{α}, α ∈ J, are continuous in T_{S} and T is the coarsest topology with this property. Thus T = T_{S}.

(c). Let g: Y → A be any map. Then

g: Y → A is continuous ⇔ ∀U ∈ S: g^{−1}(U) ∈ T_{Y}
⇔ ∀α ∈ J ∀U_{α} ∈ T_{α}: g^{−1}(f_{α}^{−1}U_{α}) ∈ T_{Y}
⇔ ∀α ∈ J ∀U_{α} ∈ T_{α}: (f_{α} ∘ g)^{−1}U_{α} ∈ T_{Y}
⇔ ∀α ∈ J: f_{α} ∘ g: Y → X_{α} is continuous
⇔f ∘ g: Y → ∏ X_{α} is continuous

where T_{Y} is the topology on Y and T_{α} the topology on X_{α}.

(d). Consider first a single map f: A → X, and give A the initial topology so that the open sets in A are the sets of the form f^{−1}U for U open in X. Then f: A → f(A) is always continuous [Thm 18.2] and open because f(A) ∩ U = f(f^{−1}U) for all (open) subsets U of X. Next, note that the initial topology for the set maps {f_{α} | α ∈ J} is the initial topology for the single map f = (f_{α}): A → ∏ X_{α}. As just observed, f: A → f(A) is continuous and open.

Example: The product topology on ∏ X_{α} is the initial topology for the set of projections π_{α}: ∏ X_{α} → X_{α}.

References