Page 160 - DMTH503_TOPOLOGY

Topology

Notes Examples of T -space
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(i) Every metric space is T -space.
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(ii) If (X, T) is cofinite topological space, then it is T -space.
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(iii) Every discrete space is T -space.
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(iv) An indiscrete space containing only one point is a T -space.
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17.1.1 T -Axiom of Separation or Frechet Space
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A topological space (X, T) is said to satisfy the T -Axiom of separation if given a pair of distinct
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points x, y  x
G, H  T s.t. x  G, y  G, y  H, x  H

In this case the space (X, T) is called T -space or Frechet space.
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Example 2: Let X = {a, b, c} with topology T = {, X, {a}, {b}, {a, b}} defined on X is not a
T -space because for a, c  X, we have open sets {a} and X such that a  {a}, c  {a}. This shows that
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we cannot find an open set which contains c but not a, so (X, T) is not a T -space. But we have
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already showed that (X, T) is a T -space. This shows that a T -space may not be a T -space. But the
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converse is always true.
Theorem 1: A topological space (X, T) is a T -space if f(x) is closed for each x  X. In a topological
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space, show that T -space  each point is a closed set.
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Proof: (i) Let (X, T) be a topological space s.t. {x} is closed  x  X.
To prove that X is T -space.
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Consider x, y  X s.t. x  y.
Then, by hypothesis, {x} and {y} are disjoint closed sets. This means that X-{x} and X-{y} are
T-open sets.
Write G = X-{y}, H = X – {x},
Then G, H  T s.t. x  G, y  G, y  H, x  H.
This proves that (X, T) is a T -space.
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(ii) Conversely, suppose that (X, T) is a T -space.
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To prove that {x} is closed  x  X.

Since X is a T -space.
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 Given a pair of distinct points x, y  X,  G, H  T.
s.t. x  G, y  G and y  H, x  H.
Evidently, G  X – {y}, H  X – {x}.
Given any x  X – {y}   G  T s.t. x  G  x – {y}.

This proves that every point x of X – {y} is an interior point of X – {y}, meaning thereby X – {y} is
open, i.e., {y} is closed. Furthermore, given any y  X – {x}   H  T s.t. y  H  X – {x}.
This implies that every point y of X – {x} is an interior point of X – {x}. Hence X – {x} is open , i.e.,

{x} is closed.

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