Let X be a non-empty set and let x 0 ∈ X. The topology T is defined by the collection of subsets U ⊂ X such that either U = ∅ or U ∋ x 0 .Is it true that if f : X → X is continuous, does it follow that f ( x 0 ) = x 0 with respect to the topology T?I under stand that the converse is true, that is for any function f : X → X with f ( x 0 ) = x 0 is continuous with respect to the topology T but i don't know how to prove if this case is true.

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Let   X be a non-empty set and let       x    0    ∈  X. The topology   T is defined by the collection of subsets   U  ⊂  X such that either   U  =  ∅ or   U  ∋      x    0  .Is it true that if   f  :  X  →  X is continuous, does it follow that   f  (      x    0    )  =      x    0   with respect to the topology   T?I under stand that the converse is true, that is for any function   f  :  X  →  X with   f  (      x    0    )  =      x    0   is continuous with respect to the topology T but i don&#039;t know how to prove if this case is true.

Xzavier Shelton · Accepted Answer

No: consider a constant map to a point that is not       x    0  .These are in fact the only counterexamples to your claim. Suppose   f  :  X  →  X is continuous. If       x    0    ∈      i    m    (  f  ) then   {      x    0    } is an open set in the codomain with non-empty preimage, and from this you can easily see that   f  (      x    0    )  =      x    0  . If       x    0    ∉      i    m    (  f  ) then it is a good exercise to show that i      i    m    (  f  ) is discrete in the subspace topology inherited from   X. But       i    m    (  f  ) must also be connected since   X is, and a connected discrete space must consist of only one point. Hence   f is constant.

Let X be a non-empty set and let x 0 </msub> ∈ X . T

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