Answer:
   
   The term "wailing wall" is not used by Jews, who instead prefer the
   term "kotel/kosel hamaaravi," Western Wall.
   
   The Western wall is a remainder of the wall on the outer perimiter of
   the mountain (although some people mistakenly believe it was the only
   remaining structure from the second temple left standing after the
   Roman destruction). Since the Jews are currently considered to be in a
   state of "ritual impurity" until a special ritual/procedure can be
   performed (notably the ashes of the red heifer), the traditional view
   is that no Jew may set foot on the actual site of the temple and this
   is the closest they can come to praying at the temple site. [Note:
   According to Rav Shlomo Goren, Jews can go on the southest 120 meters
   of the temple mount. This is the result of intersecting all the ideas
   of all the POSKIM.]

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