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Math Help - Continuity stuff

  1. #1
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    Continuity stuff

    If f: (M_1,d_1)\to(M_2,d_2) is continuous and N_1 is a vector subspace of M_1 then f\bigg|_{N_1}(N_1,d_1)\to (M_2,d_2) is continuous.

    How does one prove continuity with f restricted into a set?

    Let f:  (M_1,d_1)\to (M_2,d_2) be continuous and f(M_1)\subseteq N_2\subseteq M_2 then f: (M_1,d_1)\to(N_2,d_2) is continuous.

    I need a hint for this one.
    Thanks.
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  2. #2
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    Re: Continuity stuff

    Hey Megus.

    Can you show that the topologies have the same properties?

    For example if you start with a simple region (and no holes of any kind), then the subset of any simple region is simple and the topologies are the same then continuity in the sub-space under the preservation of the topology will have the same properties of that as the non sub-space.
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  3. #3
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    Re: Continuity stuff

    These problems are about relative topology. Given a topological space X, and a subset A, then the subspace topology for A is that a set V is open in A if and only if V = U intersect A for some U that's open in X.
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