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...Solutions to problem set connectedness let x be a and t topologies on if we say that is ner than coarser y with suppose idy continuous what the relationship between justify your claim b does in one topology imply about other c being hausdor d convergence of sequence solution by denition id only preimages open subsets are but preimage u isu itself hence map i e disconnected it can written as v where non empty thenu andv arealsoopenint isdisconnected too thus way not true consider discrete indiscrete r for instance means seperate dierent points has enough opens all so then converse false example part provides counter preserved maps net con verges converges given space dene an equivalence relation elements follows there connected subset classes called components prove yourself also described subspaces which large possible ie have property whenever compute q topological f g continu ous such every show first this either or assume former exist containing clearly latter cannot happen assumpti...