Solutions to MMA100 Topology, March 13, 2010. 1. Assume ¯A
... The empty set and X obviously satisfies the criterion of being open in X. Hence X is a topological space. The two circles considered individually as subspaces of X have the same topology as when considered as subspaces of R2 . As S 1 is⨿known to be connected, so is each of the circles as subspaces of ...
... The empty set and X obviously satisfies the criterion of being open in X. Hence X is a topological space. The two circles considered individually as subspaces of X have the same topology as when considered as subspaces of R2 . As S 1 is⨿known to be connected, so is each of the circles as subspaces of ...
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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 ...
1. The one point compactification Definition 1.1. A compactification
... which is a closed subset of X − A1 which in turn is compact. Thus X − A is itself compact and so U is in TY by definition. (3) Pick V1 , ..., Vn ∈ TY and set V = ∩ni=1 Vi . If p ∈ V then each Vi has the form Vi = Ai ∪ {p} with X − Ai being compact and closed. Therefore V = A ∪ {p} with A = ∩ni=1 Ai ...
... which is a closed subset of X − A1 which in turn is compact. Thus X − A is itself compact and so U is in TY by definition. (3) Pick V1 , ..., Vn ∈ TY and set V = ∩ni=1 Vi . If p ∈ V then each Vi has the form Vi = Ai ∪ {p} with X − Ai being compact and closed. Therefore V = A ∪ {p} with A = ∩ni=1 Ai ...
Complete Metric Spaces
... Definition: A subset D in topological space is dense it the X=D, the closure of D. Recall that the closure of a set is the smallest closed set containing it. Definition: In a metric space X a point x belongs to the interior of a set V if B(x,r) is contained in V for some r>0. Definition: A set D in ...
... Definition: A subset D in topological space is dense it the X=D, the closure of D. Recall that the closure of a set is the smallest closed set containing it. Definition: In a metric space X a point x belongs to the interior of a set V if B(x,r) is contained in V for some r>0. Definition: A set D in ...
MA4266_Lect16
... (a) If points a and b can be separated by a continuous function then they can be separated by open sets. (b) If each point x and closed set C not containing a can be separated by continuous functions then they can be separated by open sets. (c) If disjoint closed sets A and B can be separated contin ...
... (a) If points a and b can be separated by a continuous function then they can be separated by open sets. (b) If each point x and closed set C not containing a can be separated by continuous functions then they can be separated by open sets. (c) If disjoint closed sets A and B can be separated contin ...
Constructing topologies
... Comment. Using the notations above, it is immediate that the topology T Φ can also be described as the weakest topology on X, with respect to which all the maps fi : X → Yi , i ∈ I, are continuous. In the light of this remark, we will call the topology T Φ the weak topology defined by Φ. Convergence ...
... Comment. Using the notations above, it is immediate that the topology T Φ can also be described as the weakest topology on X, with respect to which all the maps fi : X → Yi , i ∈ I, are continuous. In the light of this remark, we will call the topology T Φ the weak topology defined by Φ. Convergence ...
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... A topological space X is said to be weakly countably compact (or limit point compact) if every infinite subset of X has a limit point. Every countably compact space is weakly countably compact. The converse is true in T1 spaces. A metric space is weakly countably compact if and only if it is compact ...
... A topological space X is said to be weakly countably compact (or limit point compact) if every infinite subset of X has a limit point. Every countably compact space is weakly countably compact. The converse is true in T1 spaces. A metric space is weakly countably compact if and only if it is compact ...
V.3 Quotient Space
... Suppose we have a function p : X → Y from a topological space X onto a set Y . we want to give a topology on Y so that p becomes a continuous map. Remark If we assign the indiscrete topology on Y , any function p : X → Y would be continuous. But such a topology is too trivial to be useful and the mo ...
... Suppose we have a function p : X → Y from a topological space X onto a set Y . we want to give a topology on Y so that p becomes a continuous map. Remark If we assign the indiscrete topology on Y , any function p : X → Y would be continuous. But such a topology is too trivial to be useful and the mo ...
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.