Theorem retopn

Description: The topology of the real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019)

	Ref	Expression
Assertion	retopn	⊢ ( topGen ‘ ran (,) ) = ( TopOpen ‘ ℝfld )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
2	1	tgioo2	⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ )
3		df-refld	⊢ ℝfld = ( ℂfld ↾s ℝ )
4	3 1	resstopn	⊢ ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) = ( TopOpen ‘ ℝfld )
5	2 4	eqtri	⊢ ( topGen ‘ ran (,) ) = ( TopOpen ‘ ℝfld )