Need some advice

Must my 1 month old baby wear handmitten at all times?My mother in law keep nagging to wear but her nails are short and my baby keep sweating but insist to wear hand mitten and socks and swaddle her when we are using fan only. I mean I feel hot for my baby and after feeding my baby will sweat the whole face. She was arguing to me that the hands and legs are cold but usually babies blood circulation is not good hence cannot just base on touching their hands and legs but based on checking their back to check for their body temperature.#pleasehelp