Question Number 123794 by mnjuly1970 last updated on 28/Nov/20
$$\:\:\:\:\:\:\:\:\:\:\:...{mathematical}\:\:{analysis}... \\ $$$$\:{suppose}\:\:\left({X}_{\mathrm{1}} ,\tau_{\mathrm{1}} \right)\:{and}\:\left({X}_{\mathrm{2}} ,\tau_{\mathrm{2}} \right) \\ $$$$\:{are}\:{two}\:{topological}\:{spaces}. \\ $$$${prove}\:\:{f}:\left({X}_{\mathrm{1}} ,\tau_{\mathrm{1}} \right)\rightarrow\left({X}_{\mathrm{2}} ,\tau_{\mathrm{2}} \right)\:{is}\: \\ $$$${a}\:{continuous}\:{function}\:{if}\:{only} \\ $$$${if}\:\:{for}\:{any}\:{subset}\:{A}\subseteq{X}_{\mathrm{1}} \:: \\ $$$$\:\:\:\:\:\:\:{f}\left({cl}\left({A}\right)\right)\subseteq{cl}\left({f}\left({A}\right)\right) \\ $$$$\: \\ $$