The closure of a continuous image of a closure is the closure of the image

up:: Topologically Continuous Function

Continuous functions map close inputs to close outputs.

Given a Topologically Continuous function , it can be proven that, for any set ,

Using that A continuous function’s image of the closure is a subset of the closure of its image for a Topologically Continuous Function , one knows that, for any subset ,

From this, since Closure preserves subset ordering, we have that

On the other hand, since we know that All sets are contained inside their closure, and that Function images preserve subset ordering, we have that

Since Closure preserves subset ordering, we also have that

Conclusion

Thus, for all continuous functions,

Operators on the Power Set

We can see the image of and the closure as operators over power sets, as follows

Using that, it can be observed that

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References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 30