Theorem sncldre

Description: A singleton is closed w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	sncldre	⊢ ( 𝐴 ∈ ℝ → { 𝐴 } ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )

Proof

Step	Hyp	Ref	Expression
1		rehaus	⊢ ( topGen ‘ ran (,) ) ∈ Haus
2		uniretop	⊢ ℝ = ∪ ( topGen ‘ ran (,) )
3	2	sncld	⊢ ( ( ( topGen ‘ ran (,) ) ∈ Haus ∧ 𝐴 ∈ ℝ ) → { 𝐴 } ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
4	1 3	mpan	⊢ ( 𝐴 ∈ ℝ → { 𝐴 } ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )