Cops are now investigating a mentally ill Manhattan man who threw a brick at his neighbor and then lunged at her with a knife.

The turnaround comes on the heels of a report in The Post Monday that detailed how despite the victim’s fear for her life, responding officers did next to nothing, and let Roy Rodriguez, 33, spend a night under psychiatric evaluation at a local hospital instead of arresting him.

It happened last Tuesday — and the next day he was back home despite having told his 22-year-old neighbor he wanted to “kill” her.

Now cops are finally investigating the incident in light of a Post cover story on their handling of this and other cases, the victim said Tuesday.

The Post is withholding the victim’s identity.

Cops have discretion whether to arrest or send an unstable suspect to a hospital, but sources said the officers in this case should have collared Rodriguez.

“They could still go arrest him. My question is why does it have to go to the media for them to arrest him? Do your f–k-ing job,” a police source fumed, calling the collar a “no-brainer.”

“Does he have to kill her for you to make an arrest? If he kills her, you gonna send him to the hospital? No, you’re going to arrest him.”

But, the source added, cops are hesitant to make collars, because they don’t want to get jammed up if there is a problem with the arrest — especially as Mayor de Blasio seeks to close Rikers.

The victim called 911 Tuesday saying Rodriguez ran at her with a kitchen knife while she stood in the hallway of their W. 174th St. building. Earlier, he’d thrown a brick at her, she said.

But when she called the precinct Sunday to follow up, she learned cops never took a report.

“Why wouldn’t there be a report? This is a police matter, I called 911,” said the woman, adding that cops refused to tell her why they didn’t record her complaint the first time.

“I asked that question and nobody was able to answer.”

But after The Post reported on cops’ inaction, officers asked her to come into the 33rd Precinct and give a statement — so they could make an actual police report this time, she said.

And city lawmakers are now questioning the wisdom of having beat cops make the call on hospital vs, arrest.

“While we have to consider the health of the individual, it may be safer if our policy reflects the need to detain people first and let the criminal justice system answer the questions later,” said Councilman Joe Borelli (R–Staten Island).

The NYPD and City Hall did not respond to requests for comment.