- #1

librastar

- 15

- 0

## Homework Statement

Let X be a topological space. Let A be a connected subset of X, show that the closure of A is connected.

Note: Unlike regular method, my professor wants me to prove this using an alternative route.

## Homework Equations

a) A discrete valued map, d: X -> D, is a map from a topological space X to a discrete space D.

b) A topological space X is connected if and only if every discrete valued map on X is constant.

c) Suppose X is connected and f:X -> Y is continuous and onto. Then Y is connected.

## The Attempt at a Solution

My intuition tells me that if I can use the combination of a), b) and c) I should be able to arrive at the solution. Here is my rough idea (note that cl(A) means closure):

Let A be a connected subset of X.

Let f: A -> cl(A) be a continuous and onto function.

Take a arbitrary discrete valued map d: cl(A) -> D.

Consider (composition) d[tex]\circ[/tex]f: A -> D which is a discrete valued map on X.

By b) since A is connected, d[tex]\circ[/tex]f is constant.

This shows that d is constant.

Again by b), since d is a constant, this implies that cl(A) is connected.

To me, the most troubling problem is when defining such f.

Am I allowed to simply define an f that is continuous and onto, even if I do not explicit show it is continuous and onto?

Thanks.