fɔːɹ sˈɪmɪlɚ ɹˈiːzənz ɪf ˈɛks ɪz ʌŋkˈaʊntəbəl ðˈɛn wiː hæv ɐn ɛɡzˈæmpəl wˌɛɹ ðə klˈoʊʒɚɹ əvə kəmpˈækt sˈɛt ɪz nˌɑːɾə lˈɪndɪlˌɜːf spˈeɪs.
For similar reasons if X is uncountable then we have an example where the closure of a compact set is not a Lindelöf space.