Topology/Geometry Jan 2014
... the topology induced from the standard topology on R4 , j = 1, 2. Note in particular that ∂Qj = S 1 × S 1 , j = 1, 2. Consider the quotient space X obtained by identifying (w1 , w2 ) ∈ Q1 with (w2 , w1 ) ∈ Q2 whenever w1 and w2 are both in the unit circle. Compute the fundamental group of X using th ...
... the topology induced from the standard topology on R4 , j = 1, 2. Note in particular that ∂Qj = S 1 × S 1 , j = 1, 2. Consider the quotient space X obtained by identifying (w1 , w2 ) ∈ Q1 with (w2 , w1 ) ∈ Q2 whenever w1 and w2 are both in the unit circle. Compute the fundamental group of X using th ...
TOPOLOGY PROBLEMS FEBRUARY 27, 2017—WEEK 2 1
... is continuous if and only if for every subset A ⊆ X, f [Ā] ⊆ f [A]. 7. Show that if g : X → Y and f : Y → Z are continuous, then f ◦ g : X → Z is continuous. 8. Consider the integers Z, and the rational numbers Q, as subsets of R, where the latter has the usual metric topology. (i) What is the clos ...
... is continuous if and only if for every subset A ⊆ X, f [Ā] ⊆ f [A]. 7. Show that if g : X → Y and f : Y → Z are continuous, then f ◦ g : X → Z is continuous. 8. Consider the integers Z, and the rational numbers Q, as subsets of R, where the latter has the usual metric topology. (i) What is the clos ...
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... A topological space X is said to be hyperconnected if no pair of nonempty open sets of X is disjoint (or, equivalently, if X is not the union of two proper closed sets). Hyperconnected spaces are sometimes known as irreducible sets. All hyperconnected spaces are connected, locally connected, and pse ...
... A topological space X is said to be hyperconnected if no pair of nonempty open sets of X is disjoint (or, equivalently, if X is not the union of two proper closed sets). Hyperconnected spaces are sometimes known as irreducible sets. All hyperconnected spaces are connected, locally connected, and pse ...
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.