# If X is a metric space with induced topology Ƭ, then (X,Ƭ) is Hausdorff.

NEED THIS DONE ON MICROSOFT WORD!!!!

Theorem:

If *X* is a metric space with induced topology Ƭ, then (*X*,Ƭ) is Hausdorff.

Just $7 Welcome

The contrapositive of this theorem must be true:

If (*X*,Ƭ) is not Hausdorff, then *X* is not a metric space.

1) Consider (ℝ,Ƭ) with the topology induced by the taxicab metric. Using the definition for Hausdorff, give an example of why (ℝ,Ƭ) is Hausdorff.

2) The finite complement topology on ℝ is not Hausdorff. Explain why ℝ with the finite complement topology is non-metrizable.