Is it permissible not to open the mosque for Dhuhr prayer if no one attends for the prayer and the imam prays alone?

The general principle is that mosques should be open during all prayer times for anyone present to pray therein, and they should not be closed at prayer times unless keeping them open would cause harm. The Imam must open the mosque for Dhuhr prayer when he is available, even if he prays alone, for "the resident Imam is like a congregation" in terms of virtue and ruling, and he receives the reward of congregational prayer even if he prays alone, and he does not need to repeat the prayer in another congregation.