6 | Continuous Functions
... 6.12 Note. If � : X → Y is a continuous bijection then � need not be a homeomorphism since the inverse function � −1 may be not continuous. For example, let X = {�1 � �2 } be a space with the discrete topology and let Y = {�1 � �2 } be a space with the antidiscrete topology. Let � : X → Y be given b ...
... 6.12 Note. If � : X → Y is a continuous bijection then � need not be a homeomorphism since the inverse function � −1 may be not continuous. For example, let X = {�1 � �2 } be a space with the discrete topology and let Y = {�1 � �2 } be a space with the antidiscrete topology. Let � : X → Y be given b ...
Direct limits of Hausdorff spaces
... Then the points ^i(O) ai*d j"i(i) do not have disjoint neighborhoods in Lim Xn. It is also natural to ask whether or not the conclusions of the above theorems can be strengthened; i.e., whether or not stronger separation properties will be preserved by direct limits when the bonding maps are all den ...
... Then the points ^i(O) ai*d j"i(i) do not have disjoint neighborhoods in Lim Xn. It is also natural to ask whether or not the conclusions of the above theorems can be strengthened; i.e., whether or not stronger separation properties will be preserved by direct limits when the bonding maps are all den ...
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... space Z x f(Y) is normal as the product of a normal P-space with a metrizable space. It follows that the spaces g(Y) and Yare also normal. Thus we have established that the space Z is hereditarily normal. Now we can apply Katetov's Theorem on hereditarily normal products ( [7], [11] ,..p.81!) to the ...
... space Z x f(Y) is normal as the product of a normal P-space with a metrizable space. It follows that the spaces g(Y) and Yare also normal. Thus we have established that the space Z is hereditarily normal. Now we can apply Katetov's Theorem on hereditarily normal products ( [7], [11] ,..p.81!) to the ...
11/11 := sup{|/(*)|: x £ B(X)}.
... is a Markushevich basis if [xj: j £ J] coincides with X and lin{M,: j £ 7} is weak-star dense in X*. If S is a compact topological space, C(S) is the real vector space of the continuous real-valued functions defined on S with the usual norm. S is said to be Corson if it is homeomorphic to a subspace ...
... is a Markushevich basis if [xj: j £ J] coincides with X and lin{M,: j £ 7} is weak-star dense in X*. If S is a compact topological space, C(S) is the real vector space of the continuous real-valued functions defined on S with the usual norm. S is said to be Corson if it is homeomorphic to a subspace ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.