Page 168 - DMTH503_TOPOLOGY

Topology

Notes Self Assessment

5. Show that one-to-one continuous mapping of a compact topological space onto a Hausdorff
space is a homeomorphism.
6. The product of any non-empty class of Hausdorff spaces is a Hausdorff space. Prove it.

7. Show that if (X, T) is a Hausdorff space and T* is finer than T, then (X, T*) is a T -space.
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8. Show that every finite Hausdorff space is discrete.

17.3 Summary

 T -axiom of separation:
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A topological space (X, T) is said to satisfy the T -axiom
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If for x, y  X, either  G  T s.t. x  G, y  G
or  H  T s.t. y  H, x  H
 T -axiom:
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A topological space (X, T) is said to satisfy the T -axiom if
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for x, y  X  G, H  T
s.t. x  G, y  G; y  H, x  H

 T -axiom:
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A topological space (X, T) is said to satisfy the T -axiom if for x, y  X
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 G, H  T s.t. x  G, y  H, G H = 

17.4 Keywords

Cofinite topology: Let X be a non-empty set, and let T be a collection of subsets of X whose
complements are finite along with , forms a topology on X and is called cofinite topology.

Compact: A compact space is a topological space in which every open cover has a finite sub
cover.
Discrete: Let X be any non-empty set and T be the collection of all subsets of X. Then T is called
the discrete topology on the set X.
Indiscrete space: Let X be any non-empty set and T = {X, }. Then T is called the indiscrete
topology and (X, T) is said to be an indiscrete space.
Limit point: A point x  X is said to be the limit point of A  X if each open set containing x
contains at least one point of A different from x.

17.5 Review Questions

1. Show that A finite subset of a T -space has no limit point.
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2. Prove that for any set X there exists a unique smallest T such that (X, T) is a T -space.
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3. (X, T) is a T -space iff the intersection of the nhds of an arbitrary point of X is a singleton.
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4. Show that a topological space X is a T -space iff each point of X is the intersection of all
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open sets containing it.

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