In short
There are two possible reasons for Matthew calling Jerusalem the holy city
:
Jerusalem is the location of the temple (see Isaiah 48:2, Isaiah 52:1; Revelation 11:2; Revelation 21:2, Revelation 21:10).
Jesus’ success in resisting temptation would be part of his mission to restore Jerusalem’s holiness.
Interpretation 1:
Jerusalem is called the holy city
because of the presence of the temple there (see Isaiah 48:2, Isaiah 52:1; Revelation 11:2; Revelation 21:2, Revelation 21:10).1
Interpretation 2:
Jerusalem is called the holy city in light of the nature of the temptation Jesus is experiencing in the wilderness. What was happening here, through divine intervention, would have a tremendous effect upon the center of national holiness.2
5 Then the devil took him to the holy city and set him on the pinnacle of the temple