Theorem retopon

Description: The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015)

	Ref	Expression
Assertion	retopon	⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ )

Proof

Step	Hyp	Ref	Expression
1		retop	⊢ ( topGen ‘ ran (,) ) ∈ Top
2		uniretop	⊢ ℝ = ∪ ( topGen ‘ ran (,) )
3	2	toptopon	⊢ ( ( topGen ‘ ran (,) ) ∈ Top ↔ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) )
4	1 3	mpbi	⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ )