The closure of a connected set is connected

[librastar]

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$$\circ$$f: A -> D which is a discrete valued map on X.
By b) since A is connected, d$$\circ$$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.

 

[Office_Shredder]

f isn't going to exist in general. For example, if you put the indiscrete topology on N, and look at the closure of {1}, the closure has a larger cardinality even

 

[librastar]

Ok, so I cannot just define the existence of such function.

So can you give a hint or point a direction on how to work on my problem?