haʊˈɛvɚ, sˈɪns pˈiː ɪz ˈoʊpən, ðə pɹˌiːˈɪmɪdʒ ʌv pˈiː ˌʌndɚɹ ɐ kəntˈɪnjuːəs ɪndʒˈɛkʃən fɹʌm zˈiəɹoʊ,wˈʌn wʊd biː ɐn ˈoʊpən sˈɪŋɡəl pˈɔɪnt ʌv zˈiəɹoʊ,wˈʌn, wˌɪtʃ ɪz ɐ kˌɑːntɹədˈɪkʃən.
However, since p is open, the preimage of p under a continuous injection from [0,1] would be an open single point of [0,1], which is a contradiction.