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Thread: Doubts with Functions in metric space

  1. #1
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    Doubts with Functions in metric space

    I was studying continuous functions in metric spaces and I have these doubts

    For example


    $(x,\tau)$ ; $\tau = \{x,\varnothing \} $
    $(y, \mathcal{P}(y))$


    For the funtions


    $f: x\rightarrow{y}$
    $f: y\rightarrow{x}$
    $f:x\rightarrow{x}$

    what are the conditions for the functions to be continuous?
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    Re: Doubts with Functions in metric space

    Quote Originally Posted by cristianoceli View Post
    I was studying continuous functions in metric spaces and I have these doubts
    For example
    $(x,\tau)$ ; $\tau = \{x,\varnothing \} $
    $(y, \mathcal{P}(y))$

    For the funtions
    $f: x\rightarrow{y}$
    $f: y\rightarrow{x}$
    $f:x\rightarrow{x}$
    what are the conditions for the functions to be continuous?
    From my experience there is no notation posted above that has anything to do with metric spaces.
    You need to explain the notation.
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    Re: Doubts with Functions in metric space

    Quote Originally Posted by Plato View Post
    From my experience there is no notation posted above that has anything to do with metric spaces.
    You need to explain the notation.
    Ok

    Let ${(x,\tau)}\rightarrow{(y,\mathcal{P}(y)} )$ topological space (I wrote metric space in the title but it's topological spaces) where $\tau = \{x,\varnothing \} $ and $\mathcal{P}(y)$ is the power set
    Last edited by cristianoceli; Jan 31st 2018 at 05:20 PM.
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    Re: Doubts with Functions in metric space

    Quote Originally Posted by cristianoceli View Post
    Ok
    Let ${(x,\tau)}\rightarrow{(y,\mathcal{P}(y)} )$ topological space (I wrote metric space in the title but it's topological spaces) where $\tau = \{x,\varnothing \} $ and $\mathcal{P}(y)$ is the power set
    The topology $\tau=\{X,\emptyset\}$ consists of exactly the set itself & the emptyset. It is called the trivial topology. In this case there are at most two open sets.

    The topology $\mathscr{P}(Y)$ consists of all subsets of $Y$ and is called the discrete topology for $Y$.
    In this case every subset is an open set.

    Now that are at least six different looking (but equivalent) definitions of a continuous function. You should have told us which one you are to use. The most basic is: the inverse image of an open set in $Y$ is an open set in $X$.
    So at this point, we need the definition in use here as well as exact descriptions of the functions.
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    Re: Doubts with Functions in metric space

    With those interpretations, the first, "f: X→Y, a function from the "indiscrete" topology to the "discrete" topology, since the inverse image of an open set must be open, and in the "discrete" topology, every set is open, in particular, singleton sets, the inverse image of a singleton set must be open- must be either the entire set or the empty set. That means that only the constant functions are continuous.

    The second "f: Y→X", since every set in X is open, every inverse image of any set is open so every function is continuous.

    The third is the same for the same reason- every set in X is open.
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    Re: Doubts with Functions in metric space

    Thanks!!!
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