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T0 Space A topological space X is said to be a T0-space if for any pair of distinct points of X, there exist at least one open set which contains one of them but not the other. In other words, a topological space X is said to be a T0-space if for any exist an open set such that but , there . Example: Let space, because with topology defined on X, then for a and b, there exist an open set such that and for a and c, there exist an open set such that and for b and c, there exist an open set such that and is a T0- Theorems: Let X is an indiscrete topological space with at least two points, then X is not a T0-space. Let X is a discrete topological space with at least two points, then X is not a T0-space. The real line with usual topology is T0-space. Every sub space of T0-space is T0-space. A topological space X is T0-space if and only if, for any Every two point co-finite topological space is a T0-space. Every two point co-countable topological space is a T0-space. If each singleton subset of a two point topological space is closed, then it is T0-space. If each finite subset of a two point topological space is closed, then it is a T0-space. Any homeomorphic image of a T0-space is a T0-space. A pseudo metric space is a metric space if and only if it is a T0-space. . T1 Space A topological space X is said to be a T1-space if for any pair of distinct points of X, there exist two open sets which contain one but not the other. In other words, a topological space X is said to be a T1-space if for any there exist open sets and such that and . Example: Let with topology because for , we have open sets find an open set which contains c but not a, so that true. defined on X is not a T1-space and X such that . This shows that we cannot is not a T1-space. But we have already showed is a T0-space. This shows that a T0-space may not be a T1-space. But the converse is always Example: The real line with usual topology is T1-space. To prove this, we suppose that topology on that . Further assume that consists of open intervals, so we have open sets and . This shows that the real line . Since the usual and , such with usual topology is a T1-space. Theorems: o o o o o o o o o o o Every T1-space is a T0-space An indiscrete topological space with at least two points is not a T1-space. The discrete topological space with at least two points is a T1-space. Every two point co-finite topological space is a T1-space. Every two point co-countable topological space is a T1-space. Every subspace of T1-space is T1-space. A topological space is a T1-space if and only if its each finite subsets is a closed set. Following statements about a topological space X are equivalent. (1) X ia a T1-space. (2) Each singleton subset of X is closed. (3) Each subset A of X is the intersection of its open supersets. Any homeomorphic image of a T1-space is a T1-space. If x is a limit point of a set A in a T1-space X, then every open set containing an infinite number of distinct points of A. A finite set has no limit points in a T1-space. T2 Space or Hausdorff Space A Hausdorff space or T2-space is a topological space in which each pair of distinct points can be separated by disjoint open set. In other words, a topological space x is said to be a T2-space or Hausdorff space if for any , there exist open sets and such that and . Example: Let be a non-empty set with topology discrete topology). Hence (all the subsets of X, powers set or For For For and is a T2-space. For For For Theorems: Every metric space is Hausdorff space. Every T2-space is a T1-space but the converse may not be true. Every subspace of T2-space is T2-space. In a Hausdorff space, any point and a disjoint compact subspace can be separated by open sets, in the sense that there exist disjoint open sets one contains the point and the other contains the compact subspace. Every compact subspace of Hausdorff space is closed. A one-to-one continuous mapping of a compact onto a Hausdorff space is homeomorphism. Let suppose that for all x in a dense subset D of X. Then for all x in X. A space X is Hausdorff space if and only if every point a of X is the intersection of its closed neighbourhoods. Let X be a topological space and Y a Hausdorff space. Let , from X to Y, then the set be continuous functions from a space X to a Hausdorff space Y and and be continuous function is a closed set. Regular Space Let be a topological space, then for every non-empty closed set and a point x which does not belongs to , there exist open sets and , such that and In other words, a topological space X is said to be a regular space if for any set A of X, there exist an open sets and such that and . and any closed . Example: Show that a regular space need not be a Hausdorff space. For this, let X be an indiscrete topological space, then the only non-empty closed set X, so for any , there does not exist and closed set A which does not contain x. so X is trivially a regular space. Since for any Hausdorff space. , there is only one open set X itself containing these points, so X is not a T3-Space: A regular T1-space is called a T3-space. Theorems: Every subspace of a regular space is a regular space. Every T3-space is a Hausdorff space. Let X be a topological space, then the following statements are equivalent. (1) X is a regular space. (2) For every open set U in X and a point there exist an open set Vsuch that sets. . (3) Every point of X has a local neighbourhood basis consisting of closed Completely Regular Space A topological space X is said to be completely regular space, if every closed set A in X and a point and , , then there exist a continuous function , such that . In other words, a topological space X is said to be a completely regular space if for any closed set Cnot containing x, there exist a continuous function such that and a and . Remark: Let us consider a continuous function defined as constant function is continuous therefore taking the function where and . Since , . Now And Moreover the continuous function defined in the condition for completely regular space is said to be separate the point x form the set A. Tychonoff Space: A completely regular T1-space is said to be a Tychonoff space or a -space. Note: It may be noted that since product of T1-space is T1-space and product of completely regular space is completely regular space, so product of Tychonoff space is Tychonoff space. Theorems: o o o o o o Every completely regular space is a regular space as well. Every completely regular T1-space is Hausdorff space or T2-space. Every subspace of a completely regular space is completely regular space. Product of completely regular space is a completely regular space. Every subspace of Tychonoff space is Tychonoff space. Every Tychonoff space is Hausdorff space. Normal Space Let X be a topological space and, A and B are disjoint closed subsets of X, then X is said to be normal space, if there exist open sets U and V such that , . In other words, a topological space X is said to be a normal space if for any disjoint pair of closed sets F and G, there exist open sets U and V such that , . Remarks: The collection of open sets separating the closed sets is called axiom-N. It may be noted that some topologists consider the normal space basically T1 as well, while other do not. Every discrete space containing at least two elements in a normal space. Every metric space is a normal space. T4-Space: A normal T1-space is called T4-space. Theorems: o o o Every closed subspace of normal space is normal space. A closed continuous image of a normal space is normal. A topological space is normal if and only if any closed set A and an open set Ucontaining A, o o o there is at least one open set V containing A such that . Every closed subspace of a T4-space is a T4-space. Every T1 and normal space is a regular space. If X is a normal space and f is closed continuous function from X onto a topological space Y, then Y is normal as well.