Connectedness

**Showcase_22** · March 15th 2010, 05:10 AM

Prove that any infinte set with co-finite topology is connected.

Does this question even make sense?

Let $(X,T)$ be an infinite space with co-finite topology.
Let $(U,V)$ be a two point discrete space where $U=\{0,1\}$ .

Let $f: X \rightarrow U$ be any continuous function.

I need to show that f is constant.

My big problem is this:

We know $f^{-1}(1)$ and $f^{-1}(0)$ are open.

Hence $f^{-1}(0), \ f^{-1}(1) \in T$ .

However, $X-T$ is finite so there exist finitely many points $x_1, \ldots , x_n$ where $f(x_i) \neq 0$ or $1$ where $i=1, \ldots , n$ .

By that reasoning, we have a finite number of points where the function is not defined. These points are in $X$ so the function is discontinuous at each $x_i$ .

Why is this right or wrong?

**Tinyboss** · March 15th 2010, 07:39 AM

Suppose an infinite set $X$ with cofinite topology were disconnected. Then there are open subsets $U,V$ such that $U\cap V=\varnothing$ and $U\cup V=X$ . In particular, we have that both $U$ and $X\setminus U$ are open in the co-finite topology. Is that possible?

**Showcase_22** · March 15th 2010, 07:59 AM

No! that's not possible! =D

If a set $U$ is open, it's complement should be closed! (That I do know!)

I was misinterpreting what a co-finite topology was. I didn't realise that $X \setminus U$ was also open.

**Drexel28** · March 15th 2010, 09:06 AM

Originally Posted by Showcase_22

No! that's not possible! =D

If a set $U$ is open, it's complement should be closed! (That I do know!)

I was misinterpreting what a co-finite topology was. I didn't realise that $X \setminus U$ was also open.

The cofinite topology is a neat one in the sense that it cannot support two disjoint open sets. Thus it is also, in fact, a $T_1$ space which is not Hausdorff. Also, it is clear that since every totally disconnected space is Hausdorff that a cofinite space is also not totally disconnected (this was immediately obvious before actually doing the question was my point)

A more interesting question you may want to ask yourself: is a cofinite space every locally connected?

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