There are numerous mysteries surrounding the character Libby on Lost, but unfortunately, the show's creators have indicated that her story on Lost is completed. To this I say, no it's not.
Libby was part of the tail section survivors, who we met in season 2. She claimed to be a clinical psychologist and on the island, she and Hurley formed a relationship. She was "accidentally" shot by Michael and later died on the island.
The biggest mystery with Libby is why was she in Santa Rosa Mental Health Institute. We see her at the end of "Dave" sitting near Hurley. But when Hurley thought that he knew her from somewhere, she denied that they had ever met.
The other big plot point involving Libby is that she provided Desmond with the sailboat that he used in the race around the world. The "Elizabeth" was her late husband David's boat and after initially turning down her offer, Desmond uses the boat and eventually ends up on the island. ("Live Together, Die Alone").
After Libby was killed on the island, we've seen her in Michael's visions but it doesn't look like she'll be back on the show. We don't know her last name, don't why she was flying from Sydney, don't know why she was institutionalized, don't know who her husband was, or why she would give the boat to a total stranger in Desmond (or did she somehow know him?).
There are many theories about her, including that she worked for Charles Widmore (to set up Desmond). She is one of the other people who the Oceanic 6 claim to have survived the crash, but later died. There may not be any significance to this, but it was probably Hurley's way of remembering her.
The A Team-- 2015 Walk Details
2 years ago