Asplund spaces for beginners
... a dense set. This is not quite the standard definition, but we will soon see that it is equivalent. A problem for newcomers to this area is that there is a plethora of properties equivalent to Asplundness; finding a reasonable proof of the particular implication that one needs may involve a wild goo ...
... a dense set. This is not quite the standard definition, but we will soon see that it is equivalent. A problem for newcomers to this area is that there is a plethora of properties equivalent to Asplundness; finding a reasonable proof of the particular implication that one needs may involve a wild goo ...
SAM III General Topology
... To a topological space X = (X , τ ) one associates a category Π1 X defined as follows: objects of the category are points in the topological space, while a morphism from x to y is a homotopy class [f ] of a path f from x to y ; composition of morphisms is defined by the formula [g ] ◦ [f ] = [g ◦ f ...
... To a topological space X = (X , τ ) one associates a category Π1 X defined as follows: objects of the category are points in the topological space, while a morphism from x to y is a homotopy class [f ] of a path f from x to y ; composition of morphisms is defined by the formula [g ] ◦ [f ] = [g ◦ f ...
Monotone meta-Lindelöf spaces
... the set of all irrational numbers and the set of all integers respectively. The spaces [0, ω1 ) and [0, ω1 ] are the usual ordinal spaces unless specifically stated and the space [0, 1] is the subspace of the real line R. Other terms and symbols can be found in [6] and [10]. 2. The definition of mono ...
... the set of all irrational numbers and the set of all integers respectively. The spaces [0, ω1 ) and [0, ω1 ] are the usual ordinal spaces unless specifically stated and the space [0, 1] is the subspace of the real line R. Other terms and symbols can be found in [6] and [10]. 2. The definition of mono ...
General Topology - Faculty of Physics University of Warsaw
... Let X, Y be topological spaces and f : X → Y . Theorem 1.9 Let x0 ∈ X. TFAE: (1) For any neighborhood V of f (x0 ), f −1 (V ) is a neighborhood of x0 ; (2) There exists a basis Vx0 of neighborhoods of x0 and a basis of neighborhoods Wf (x0 ) of f (x0 ) such that for any W ∈ Wf (x0 ) there exists V ∈ ...
... Let X, Y be topological spaces and f : X → Y . Theorem 1.9 Let x0 ∈ X. TFAE: (1) For any neighborhood V of f (x0 ), f −1 (V ) is a neighborhood of x0 ; (2) There exists a basis Vx0 of neighborhoods of x0 and a basis of neighborhoods Wf (x0 ) of f (x0 ) such that for any W ∈ Wf (x0 ) there exists V ∈ ...
Categories of certain minimal topological spaces
... topologies defined on denumerable spaces. Also, it will be shown that the former result can be extended to all minimal regular spaces. THEOREM 3. (i) Every countably infinite minimal Frichet space is of first category; (ii) every uncountably infinite minimal Frichet space is of second category. PROO ...
... topologies defined on denumerable spaces. Also, it will be shown that the former result can be extended to all minimal regular spaces. THEOREM 3. (i) Every countably infinite minimal Frichet space is of first category; (ii) every uncountably infinite minimal Frichet space is of second category. PROO ...
