Swimming in a Pool while His Father is Sitting Shiva

For someone whose father is observing Shiva after the passing of his father, the law states that until the Sabbath following the burial, it is forbidden for him to bathe in hot water in the city where his grandfather is being mourned. However, it is permissible to bathe in lukewarm water. Is it permissible to go to a swimming pool?

Even though it is permissible to wash one's entire body in lukewarm water, it is forbidden to immerse one's entire body in a bathhouse with lukewarm water. Therefore, one should not go to a swimming pool during these days.

In 'Chochmat Adam' chapter 161, paragraph 5, it is written that the custom is to bathe only in completely cool water. Afterwards, it is written that the custom is not to go to the bathhouse, implying by this that even though it is permitted to bathe, it is forbidden to go to a bathhouse.