Theorem. Continuity of split [dt-0024]

For continuous functions f and g, the split function \langle f, g \rangle is continuous.

Proof.

Using the alternative characterisation of continuity:

  • \langle f, g \rangle is monotonic. Let x \sqsubseteq y \in A. Then: \begin {array}{lclr} \langle f , g \rangle \ x & = & (f(x), g(x)) & \text {(def)}\\ & \sqsubseteq & (f(y), g(y)) & \text {(monotonicity of $f$,$g$)}\\ & = & \langle f , g \rangle \ y & \text {(def)} \end {array}
  • \langle f , g \rangle preserves lubs of directed sets. Let X \subseteq A be directed. Then:
    • \begin {array}{lclr} \langle f , g \rangle \ (\bigsqcup X) & = & (f(\bigsqcup X), g(\bigsqcup X)) & \text {(def)}\\ & = & (\bigsqcup \{ f(x) \mid x \in X \}, \bigsqcup \{ g(x) \mid x \in X \}) & \text {(continuity of $f$,$g$)}\\ & = & \bigsqcup \{ \langle f , g \rangle \ x \mid x \in X \} & \text {(def)} \end {array}