Product spaces
... The converse of this is false, i.e. a subset W of X × Y that contains (p, q) may not be open. An example is any set W = {(p, q), (x, y)} with x 6= p and y 6= q. Thus, the product topology of Tp and Tq on R2 is not equal to the included point topology T(p,q) on R2 . The result of the following lemma ...
... The converse of this is false, i.e. a subset W of X × Y that contains (p, q) may not be open. An example is any set W = {(p, q), (x, y)} with x 6= p and y 6= q. Thus, the product topology of Tp and Tq on R2 is not equal to the included point topology T(p,q) on R2 . The result of the following lemma ...
CW complexes
... q −1 are continuous since p and π are both identification maps and q ◦ π = p and q −1 ◦ p = π are continuous. Thus q : X/∼ → Y is a homeomorphism so from a topological point of view we consider Y and X/∼ to be the same space. Thus quotient spaces and identification spaces are one and the same thing. ...
... q −1 are continuous since p and π are both identification maps and q ◦ π = p and q −1 ◦ p = π are continuous. Thus q : X/∼ → Y is a homeomorphism so from a topological point of view we consider Y and X/∼ to be the same space. Thus quotient spaces and identification spaces are one and the same thing. ...
THE INTERSECTION OF TOPOLOGICAL AND METRIC SPACES
... are already defined and satisfy (1). Note that O m−1 and X \ O m+1 2n 2n 2n 2n are disjoint closed sets. Using normality, and the same arguments used for defined O 21 we now pick O 2mn satisfying our wanted properties. Thus we have defined these sets for all elements of D. We now move on to defining ...
... are already defined and satisfy (1). Note that O m−1 and X \ O m+1 2n 2n 2n 2n are disjoint closed sets. Using normality, and the same arguments used for defined O 21 we now pick O 2mn satisfying our wanted properties. Thus we have defined these sets for all elements of D. We now move on to defining ...
Seminar in Topology and Actions of Groups. Topological Groups
... said to be compatible if they satisfy (i) and (ii). Example 1. 1.The discrete topology on a group G is compatible with the group structure. A topological group whose topology is discrete is called a discrete group. 2. The trivial topology on G is compatible with the group structure of G. 3. Every no ...
... said to be compatible if they satisfy (i) and (ii). Example 1. 1.The discrete topology on a group G is compatible with the group structure. A topological group whose topology is discrete is called a discrete group. 2. The trivial topology on G is compatible with the group structure of G. 3. Every no ...
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.