First of all , you should invest in some stable memes to get a bit of experience then you can gamble and risk on not so popular memes . I recommend , if you wanna be adventorous to start with Biggie Cheese , it ' s a fresh new meme and it ' s rising very fast . I will invest in it very soon . <| sep |> Swissvale is so much gayer than everyone knows lol <| sep |> As others have pointed out here , it is trivial that this bijection cannot be continuous w . r . t to the usual topologies on both sets . It would be interesting though to ask under which topologies such a map could be continuous . Either we need to make sure 0 , 1 isn ' t compact , or make R compact . Though I ' m guessing the question here is also fairly trivial to someone who has more background in topology than I do , as it is simply asking the question under which topologies the two spaces are homeomorphic .