Theorem kgenftop

Description: The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015)

	Ref	Expression
Assertion	kgenftop	$$⊢J\in \mathrm{Top}\to \mathrm{𝑘Gen}\left(J\right)\in \mathrm{Top}$$

Proof

Step	Hyp	Ref	Expression
1		toptopon2	$$⊢J\in \mathrm{Top}\leftrightarrow J\in \mathrm{TopOn}\left(\bigcup J\right)$$
2		kgentopon	$$⊢J\in \mathrm{TopOn}\left(\bigcup J\right)\to \mathrm{𝑘Gen}\left(J\right)\in \mathrm{TopOn}\left(\bigcup J\right)$$
3	1 2	sylbi	$$⊢J\in \mathrm{Top}\to \mathrm{𝑘Gen}\left(J\right)\in \mathrm{TopOn}\left(\bigcup J\right)$$
4		topontop	$$⊢\mathrm{𝑘Gen}\left(J\right)\in \mathrm{TopOn}\left(\bigcup J\right)\to \mathrm{𝑘Gen}\left(J\right)\in \mathrm{Top}$$
5	3 4	syl	$$⊢J\in \mathrm{Top}\to \mathrm{𝑘Gen}\left(J\right)\in \mathrm{Top}$$