Topologically Continuous Function

up:: Topological Space

A function $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ mapping between topological spaces is said to be continuous if

$$\forall A_Y\in {\tau }_Y,f^{-1}(A)\in {\tau }_X$$

That is, it “premaps” open sets (from $Y$) to open sets (in $X$).

Relation to Metric Space continuous functions

Corollaries

It can also be proven that Continuous functions premap closed sets to closed sets.

A more motivated equivalent definition of continuity is that in which A continuous function’s image of the closure is a subset of the closure of its image, through which one can easily see that Continuous functions map close inputs to close outputs.

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References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 27