Theorem ordttop

Description: The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015)

	Ref	Expression
Assertion	ordttop	$$⊢R\in V\to \mathrm{ordTop}\left(R\right)\in \mathrm{Top}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\mathrm{dom}R=\mathrm{dom}R$$
2	1	ordttopon	$$⊢R\in V\to \mathrm{ordTop}\left(R\right)\in \mathrm{TopOn}\left(\mathrm{dom}R\right)$$
3		topontop	$$⊢\mathrm{ordTop}\left(R\right)\in \mathrm{TopOn}\left(\mathrm{dom}R\right)\to \mathrm{ordTop}\left(R\right)\in \mathrm{Top}$$
4	2 3	syl	$$⊢R\in V\to \mathrm{ordTop}\left(R\right)\in \mathrm{Top}$$