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Subspace vs. subset

  1. Feb 10, 2009 #1
    Hey guys....

    I'm not sure how i'm suppose to show that if Y is a subspace of X, and A is a subset of Y, then the topology A inherits as a subspace of Y is the same as the topology it inherits as a subspace of X.

    I know that a subspace is... Ty = {Y[tex]\cap[/tex]U| U [tex]\in[/tex]T}
    meaning that its open sets consist of all intersections of open sets of X with Y.
    and that a subset is every element of A is also an element of B.

    pretty much right? so how do i express this in terms of subset and subspace?
     
  2. jcsd
  3. Feb 10, 2009 #2

    CompuChip

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    Let TY denote the topology inherited from Y, and TX the topology inherited from X, i.e.
    [tex]T_Y = \{ U \cap A | U \text{ is open in } Y \} [/tex]
    and
    [tex]T_X = \{ V \cap A | V \text{ is open in } X \} [/tex]

    First let's show that [itex]T_Y \subseteq T_X[/itex]. Let [itex]U \in T_Y[/itex] be an open set in the Y-induced topology on A. That means there is some open set U' in Y, such that [itex]U = A \cap U'[/itex]. Can you find a set U'' which is open in X, such that [itex]U = A \cap U''[/itex]? Because that would show that
    [tex]U \in T_Y \implies U \in T_X[/tex]
    and therefore
    [tex]T_Y \subseteq T_X[/tex].
     
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