Irreducible Space is Connected

Theorem

Let $T = \struct {S, \tau}$ be a topological space which is irreducible.


Then $T$ is connected.


Proof

Let $T = \struct {S, \tau}$ be irreducible.

Then:

$\forall U_1, U_2 \in \tau: U_1, U_2 \ne \O \implies U_1 \cap U_2 \ne \O$

So trivially there are no two open sets that can form a separation of $T$.

The result follows from definition of connected.

$\blacksquare$


Also see


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.