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By A. F. Beardon

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There is with respect to which, f is continuous: this is defined by B e Tf if and only if It is easy to check that T^ f 1 (B) e T. is a topology, that that this is the largest topology which makes the quotient topology induced by f f is continuous and continuous. We call T^ f. An important example of a quotient topology arises from a group G of homeomorphisms of a space in X X onto itself. The orbit of a point x is G (x) = {g (x) : g € G} and the space of orbits is denoted by f : X that X/G X/G taking each x X/G.

This is because any union of sets of type (1) is also of type (1) 32 and similarly for (2): the latter because if each intersection riK^ is compact, then the is closed (being an intersection of closed sets) and so is compact. This shows that the induced topology on X is the given topology * As X is compact (a neighbourhood of 00 omits to cover only a compact * set) we see that X is a compactification of X. * To show that each x in X is Hausdorff, it is only necessary to separate X from » by open sets.

Deduce that X is connected but not arcwise connected. 7 QUOTIENT SPACES Let If we give Y T be a topology on a topology, then a largest topology T^ on Y f X and let f : X Y be any function. may or may not be continuous. There is with respect to which, f is continuous: this is defined by B e Tf if and only if It is easy to check that T^ f 1 (B) e T. is a topology, that that this is the largest topology which makes the quotient topology induced by f f is continuous and continuous. We call T^ f. An important example of a quotient topology arises from a group G of homeomorphisms of a space in X X onto itself.

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