This question originally appeared on Quora. 
By Jennifer Veloso, Harry Potter Fan
The chamber was a legend way before it was opened by Tom Riddle Jr., back from when Slytherin first left Hogwarts. Nobody believed it actually existed until the events leading up to Hagrid's expelling, and even then, they thought it just coincided with the legend, not confirmed it. Hagrid was expelled because they thought his pet creature (Aragog) killed Myrtle, not because they thought he opened the chamber and Aragog was the beast of legend.
Once Aragog was exposed, they assumed they had found the creature behind the attack, so the official explanation was that it was an unintended consequence of a bad decision by Hagrid (which, incidentally, could have been true ... after all, Hagrid almost got Harry and Ron killed in the forest because he trusted Aragog too much). Even when they DID try searching for evidence that a chamber existed, nothing was found.
Out of respect for Hagrid's privacy, though, people don't bring up this information about him. This might be why the legend persisted (with more vigor) if the students - and later, as the students grew up and had children, future generations - were not completely clear on the events that unfolded.
I think it's implied that Dumbledore knew it wasn't Hagrid's (i.e. Aragog's) doing, though. Remember, he kept a "close watch" on Riddle after those events. Why then would he not seek out the chamber until it was found? Perhaps he did not think he would convince others that the chamber existed, and accusing Tom (at the time seen as a respectable, responsible, bright student) without any proof would be both unfair and not be met with favorable response. And he might have weighed that if the chamber did exist and only Slytherin's heir could open it, it is unlikely to be opened again once the culprit (of whose identity he had a strong suspicion) left school.
Maybe not the best decision to let the chamber matter settle itself, but as we came to know well Dumbledore's mistakes often had bigger repercussions than the average person's.
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