Firefighters were called to a blaze amid reports that people were still inside the property on The Spinney in Lancaster at around 5am on March 31.
Crews arrived to find the ground floor well alight with the family already outside the two-storey house.
Four adults and child were treated for smoke inhalation at the scene and were taken to Royal Lancaster Infirmary by ambulance.
A 49-year-old woman remains in hospital suffering from smoke inhalation. Her condition is reported to be stable, police say.
The child, a three-year-old boy who is described as "poorly but not critical", has been transferred to Alder Hey Children's Hospital.
A representative for the fire service said: "Three fire engines and crews (two from Lancaster, one from Morecambe) responded to the fire call.
"The ground floor of the two-storey house was well alight but thankfully no-one was trapped inside and all people were accounted for.
"Firefighters gave oxygen to occupants who had breathed in smoke. The fire is believed to have been started deliberately."
DI Phil Jones from Lancaster CID said: “We are appealing to anyone who may have witnessed what happened or who has any information about the fire and has not yet spoken to us to get in touch.”
Anyone with information can contact police on 101 quoting log 165 of March 31st. Alternatively Crimestoppers can be contacted anonymously on 0800 555 111 or online at Crimestoppers-uk.org.