Theorem conntop

Description: A connected topology is a topology. (Contributed by FL, 22-Dec-2008) (Revised by Mario Carneiro, 14-Dec-2013)

	Ref	Expression
Assertion	conntop	$$⊢J\in \mathrm{Conn}\to J\in \mathrm{Top}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\bigcup J=\bigcup J$$
2	1	isconn	$$⊢J\in \mathrm{Conn}\leftrightarrow \left(J\in \mathrm{Top}\wedge J\cap \mathrm{Clsd}\left(J\right)=\left\{\varnothing ,\bigcup J\right\}\right)$$
3	2	simplbi	$$⊢J\in \mathrm{Conn}\to J\in \mathrm{Top}$$