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*General Topology*:

*If [tex]X[/tex] and [tex]Y[/tex] are topological spaces and [tex]f:X\to Y[/tex] is continuous and either open or closed, then the topology [tex]\tau[/tex] on [tex]Y[/tex] is the quotient topology induced by [tex]f[/tex].*

So [tex]f[/tex] has to be onto doesn't it? Otherwise there will be multiple topologies on [tex]Y[/tex] that satisfy the hypotheses but are not the quotient topology?