4. Topologies and Continuous Maps.
... Remark. Note that the concept of subbasis makes it now a trivial matter to construct topologies for a set X. For this we simply give ourselves any collection A of subsets of X whose union is X. This already defines a unique topology, given by first taking all intersections of elements of A and then ...
... Remark. Note that the concept of subbasis makes it now a trivial matter to construct topologies for a set X. For this we simply give ourselves any collection A of subsets of X whose union is X. This already defines a unique topology, given by first taking all intersections of elements of A and then ...
Topology Proceedings 6 (1981) pp. 329
... If g and f are real-valued functions defined on a topological space X such that g < f (i.e., g(x) < f(x) for each x in X), consider the problem of finding necessary and sufficient conditions in order for there to be a continuous function h defined on X such that g < h < f. ...
... If g and f are real-valued functions defined on a topological space X such that g < f (i.e., g(x) < f(x) for each x in X), consider the problem of finding necessary and sufficient conditions in order for there to be a continuous function h defined on X such that g < h < f. ...
Murat D_iker, Ankara
... 2.2. Example. Consider the set X = {(x, y): x > 0, y > 0} ⊂ R 2 . Let u consist of ∅ and all subsets G of X satisfying (i) (x, y) ∈ G, 0 < x 6 x ⇒ (x , y) ∈ G (ii) (x, y) ∈ G, 0 < y 6 y ⇒ (x, y ) ∈ G (iii) ∃y > 0 with (0, y) ∈ G. Clearly, u is a topology on X , and so is v = {G 1 : G ∈ u}. The space ...
... 2.2. Example. Consider the set X = {(x, y): x > 0, y > 0} ⊂ R 2 . Let u consist of ∅ and all subsets G of X satisfying (i) (x, y) ∈ G, 0 < x 6 x ⇒ (x , y) ∈ G (ii) (x, y) ∈ G, 0 < y 6 y ⇒ (x, y ) ∈ G (iii) ∃y > 0 with (0, y) ∈ G. Clearly, u is a topology on X , and so is v = {G 1 : G ∈ u}. The space ...
On Almost T -m- continuous Multifunctions
... multifunctionsand their properties are studied by ValeriuPopa, and Takashi Noiri[7],[9],[10], Whileat 2006,A. Kanibir and I.L. Reilly [2]investigate the conceptof Almost -continuous multifunctions. After thatHadiJaber Mustafa and Muayad G. Mohsen [6] introduced a stronger concept than almost -contin ...
... multifunctionsand their properties are studied by ValeriuPopa, and Takashi Noiri[7],[9],[10], Whileat 2006,A. Kanibir and I.L. Reilly [2]investigate the conceptof Almost -continuous multifunctions. After thatHadiJaber Mustafa and Muayad G. Mohsen [6] introduced a stronger concept than almost -contin ...
Math 190: Quotient Topology Supplement 1. Introduction The
... Step 3 typically requires the most ingenuity – the Pasting Lemma is not infrequently applied here. Probably the most useful fact is that the quotient map X → (X/ ∼) is continuous. Step 4 is another set-theoretic check that the candidate inverse maps are actually inverse to one another. 6. Quotient M ...
... Step 3 typically requires the most ingenuity – the Pasting Lemma is not infrequently applied here. Probably the most useful fact is that the quotient map X → (X/ ∼) is continuous. Step 4 is another set-theoretic check that the candidate inverse maps are actually inverse to one another. 6. Quotient M ...
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... A topological space X is connected im kleinen at a point x if every open set U containing x contains an open set V containing x such that if y is a point of V , then there is a connected subset of U containing {x, y}. Another way to say this is that X is connected im kleinen at a point x if x has a ...
... A topological space X is connected im kleinen at a point x if every open set U containing x contains an open set V containing x such that if y is a point of V , then there is a connected subset of U containing {x, y}. Another way to say this is that X is connected im kleinen at a point x if x has a ...
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... two topological spaces X and Y and a continuous map f : X → Y , can one infer that one of the spaces has a certain topological property from the fact that the other space has this property? A trivial case of this question may be disposed of. If f is a homeomorphism, then the spaces X and Y cannot be ...
... two topological spaces X and Y and a continuous map f : X → Y , can one infer that one of the spaces has a certain topological property from the fact that the other space has this property? A trivial case of this question may be disposed of. If f is a homeomorphism, then the spaces X and Y cannot be ...
