SMSTC (2014/15) Geometry and Topology www.smstc.ac.uk
... Examples 1.13 Any set X can be given the discrete topology, in which every subset is open, or the indiscrete topology, in which the only open sets are X and ∅. Example 1.14 Given a subset A ⊂ X of a topological space X, the subspace topology on A is formed by taking V ⊂ A to be open if and only if V ...
... Examples 1.13 Any set X can be given the discrete topology, in which every subset is open, or the indiscrete topology, in which the only open sets are X and ∅. Example 1.14 Given a subset A ⊂ X of a topological space X, the subspace topology on A is formed by taking V ⊂ A to be open if and only if V ...
![arXiv:math/0204134v1 [math.GN] 10 Apr 2002](http://s1.studyres.com/store/data/000969919_1-7ee0f69619dac66dbe1138932726f1da-300x300.png)
Alexandrov one-point compactification
... space X is obtained by adjoining a new point ∞ and defining the topology on X ∪ {∞} to consist of the open sets of X together with the sets of the form U ∪ {∞}, where U is an open subset of X with compact complement. With this topology, X ∪{∞} is always compact. Furthermore, it is Hausdorff if and o ...
... space X is obtained by adjoining a new point ∞ and defining the topology on X ∪ {∞} to consist of the open sets of X together with the sets of the form U ∪ {∞}, where U is an open subset of X with compact complement. With this topology, X ∪{∞} is always compact. Furthermore, it is Hausdorff if and o ...
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... at x if there is a sequence (Bn )n∈N of open sets such that whenever U is an open set containing x, there is n ∈ N such that x ∈ Bn ⊆ U . The space X is said to be first countable if for every x ∈ X, X is first countable at x. Remark. Equivalently, one can take each Bn in the sequence to be open nei ...
... at x if there is a sequence (Bn )n∈N of open sets such that whenever U is an open set containing x, there is n ∈ N such that x ∈ Bn ⊆ U . The space X is said to be first countable if for every x ∈ X, X is first countable at x. Remark. Equivalently, one can take each Bn in the sequence to be open nei ...
Topology, MM8002/SF2721, Spring 2017. Exercise set 4 Exercise 1
... Exercise 1. Let X be topological space, Y be a set and f : X → Y be a surjective map. Recall that a subset U ⊆ Y is open in the quotient topology, if and only if f −1 (U ) is open in X. • Show that the quotient topology is in fact a topology. • Show that the quotient topology is the finest topology ...
... Exercise 1. Let X be topological space, Y be a set and f : X → Y be a surjective map. Recall that a subset U ⊆ Y is open in the quotient topology, if and only if f −1 (U ) is open in X. • Show that the quotient topology is in fact a topology. • Show that the quotient topology is the finest topology ...
Handout 1
... about the Klein bottle as two Möbius bands glued together along their boundary circles. (6) Let X = I 2 be the unit square with the equivalence relation, (t, 0) ∼ (1 − t, 1) and (0, t) ∼ (1, 1 − t) for all 0 6 t 6 1, gluing the opposite sides in pairs and reversing the orientation of both pairs. Th ...
... about the Klein bottle as two Möbius bands glued together along their boundary circles. (6) Let X = I 2 be the unit square with the equivalence relation, (t, 0) ∼ (1 − t, 1) and (0, t) ∼ (1, 1 − t) for all 0 6 t 6 1, gluing the opposite sides in pairs and reversing the orientation of both pairs. Th ...
