By Milgram R. (ed.)
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Extra resources for Algebraic and Geometric Topology, Part 1
18 suggest (correctly) that subspaces are usually easy to handle because the ‘structure’ just gets traced or shadowed onto the subset that carries the subspace topology, there is also a rich source of errors here. Suppose that we are given (X, 3) and B » A » X. If we say ‘B is open’, then there are two distinct things that we might mean: that B is a 3-open subset of X which happens to be contained in A, or that B is a 3A -open set. It is vital to realise that these are diﬀerent; similar remarks apply to ‘closed’, to ‘neighbourhood’, to ‘closure’ and so on.
N is a neighbourhood of x K x g N. N is a neighbourhood of x K x g N and X \ N is ﬁnite. N is a neighbourhood of x K x g N and X \ N is countable. N is a neighbourhood of x K p g N and x g N. N is a neighbourhood of x K ⎧ ⎪ ⎨x = p & N = X or ⎪ ⎩ x = p & either (x g N and p g N) or N = X. 5 Let N1 , N2 , . . , Nj be ﬁnitely many neighbourhoods of p (in X). Choose open sets G1 , G2 , . . , Gj so that p g G1 » N1 , p g G2 » N2 , . . , p g Gj » Nj . j j j 1 1 Then | Gi is open, and p g | Gi » | Ni 1 j | Ni 1 therefore is a neighbourhood of p.
31 32 3 CONTINUITY AND CONVERGENCE (ii) A mapping f : (X, 3) C (Y, 3 ) is called closed if K is 3-closed I f (K) is 3 -closed, that is, if the direct image of every closed set is closed. 13 Examples (i) Any map into a discrete space is both an open map and a closed map. Any onto map with trivial domain space is both an open map and a closed map. (ii) Any onto map from (R, 3cf ) to (R, 3cc ) is a closed map. 14 Proposition For a one-to-one and onto mapping f : (X, 3) C (Y, 3 ), the following are equivalent: (i) f is continuous, (ii) f –1 is an open map, (iii) f –1 is a closed map.