The Tenth of Teves is a fast partly because it is the beginning of the siege that led to the walls of Jerusalem being breached.
While this is a negative event, there are positive aspects to this tragedy as well, for it signifies an opening in the walls. Openings to the outside, allow one to draw the outside into himself, purify and uplift it. It also allows one to expel the negative from the inside out.
In Chassidus, words of Torah and prayer are compared to bricks/stones.
There is a parable told about two people praying. One was asked why he davens so slow?
He replied that he builds a wall with the fence posts close together so that nothing foreign can penetrate and he can, therefore, concentrate and spend time praying. The other person replied that he davens fast so the posts are far apart and his own unworthy thoughts can be ejected. Thus, openings and breaches, which at first glance are an unmitigated evil, can also serve as the means to expel that which is unworthy.
Thus a person has a wall which, like a wall of a cell in his body, is a semi-permeable membrane. Man's borders (made famous by the book by Rabbi Manis Friedman) are local at the same time as they are global, for his walls are the walls of a sancified space in his own Temple.