Open subsets of a irreducible topological space

Question

I have this problem:

"Show that a topological space X$\neq\emptyset$ is irreducible $\Leftrightarrow$ $\forall$ U open subset of X, U is connected. "

I can easily prove the $\Rightarrow$ part, using that every open subset of an irreducible space is itself irreducible (therefore connected).

But what about the $\Leftarrow$ part? I have no idea how to prove it, and to be fair I'm not even sure it's true.

Thanks everybody!

Does this answer your question? An open subset of an irreducible set is dense. — Jacopo Lanzoni, Dec 07 '22 at 19:59

score 3 · Answer 1 · answered Mar 30 '18 at 14:34

3

Let $A$, $B$ be any two non-empty open sets in $X$. Then, $A\cup B$ is open and non-empty, and thus connected. So $A\cap B\ne\emptyset$.

answered Mar 30 '18 at 14:34

Does that mean, that the other direction is valid as well ? – Peter Mar 30 '18 at 14:35
1

@Peter Indeed. Irreducible $\Leftrightarrow$ all open are connected $\Leftrightarrow$ all non-empty open are dense. – Mar 30 '18 at 14:36
OK, I have overseen the "for all" – Peter Mar 30 '18 at 14:37

Henno Brandsma · Accepted Answer · 2018-03-30T22:01:59.790

Suppose $U$ is non-empty open and disconnected so that $U =U_1 \cup U_2$, where $U_1$ and $U_2$ are non-empty disjoint and open in $U$ (and so also open in $X$).

Then $X = (X\setminus U_1) \cup (X\setminus U_2)$ (as $U_1$ and $U_2$ are disjoint) and both sets are closed and proper (as both $U_i$ are non-empty). This contradicts that $X$ is irredicible.

So all non-empty open $U$ are connected.

The other direction is similar. Suppose $X$ has no open disconnected subsets and suppose that $X = F_1 \cup F_2$ where $F_1$ and $F_2$ are proper non-empty closed sets. Then (by de Morgan) $$\emptyset = (X\setminus F_1) \cap (X\setminus F_2)$$

so $$U = (X\setminus F_1) \cup (X\setminus F_2)$$ is an open set that is disconnected (as both sets in this union are non-empty proper open subsets of $U$). This contradiction shows that $X$ is irreducible.

Open subsets of a irreducible topological space

2 Answers2