Gardaí said the family are "appreciative and overwhelmed" by the national outpouring of support.
The family of Ashling Murphy has requested "privacy, space and time" to process the 23-year-old's death.
It comes as vigils for the primary school teacher who was attacked and killed while out jogging along the canal bank at Cappincur in Tullamore, Offaly at around 4pm on Wednesday continue across the country.
Earlier on Friday, it was reported that a
"person of interest" has been identified by Gardaí investigating the murder.
According to
RTÉ, the authorities are waiting to speak with a man who is currently in hospital.
While Gardaí said in a statement that "significant progress" has been made in the investigation to date, they stated they would not be confirming any specific details for operational reasons.
"An Garda Síochána continue to support Ashling's family at this time," a Garda spokesperson said.
"The Murphy family are appreciative and overwhelmed by the national outpouring of support shown to them.
"The Murphy family have requested that they now need privacy, space and time to process Ashling's death."
Gardaí also said they wished to acknowledge the support of the public to date.
They added that in order to assist the public in providing information to the investigation team, the dedicated phone line 057 9357060 has been established at Tullamore Garda station.
"An Garda Síochána continue to appeal to any person who has not made contact with the investigation team and who has any information on this fatal assault to come forward," the spokesperson said.
"An Garda Síochána continue to appeal for any information on a Falcon Storm mountain bike with straight handlebars and distinctive yellow/green front forks prior to 4pm on Wednesday, 12 January 2022 and in the previous days/weeks.
Image via An Garda Síochána
"Gardaí continue to appeal to any persons that were in the Cappincur/Canal walk area of Tullamore, County Offaly before 4pm, Wednesday, 12 January 2022, to make contact with them.
"Gardaí continue to appeal to any person in and between the Tullamore town centre area and the wider approach roads to the Grand Canal Way in the vicinity of Capincur on the 12 January 2022, with any form of video footage (dashcam, CCTV, Mobile phone, GoPro cam etc..) to contact us."
Anyone with information is asked to contact Tullamore Garda Station on 057 9357060, the Garda Confidential Line on 1800 666 111 or any Garda Station.
Also on Friday, President Michael D Higgins paid tribute to Ashling Murphy, saying that
"she represented the best of her generation" and that her life will be recalled as "inspirational".
Main image via Eamonn Farrell / RollingNews.ie
