Functions
... A relation can be a relationship between sets of information. For example, consider the set of all of the people in your Algebra class and the set of their heights is a relation. The pairing of a person and his or her height is a relation. In relations and functions, the pairs of names and height ar ...
... A relation can be a relationship between sets of information. For example, consider the set of all of the people in your Algebra class and the set of their heights is a relation. The pairing of a person and his or her height is a relation. In relations and functions, the pairs of names and height ar ...
Building new topological spaces through canonical maps
... Suppose (X, TX ) and (Y, TY ) are topological spaces. (In other words, X and Y are sets, and TX and TY are subsets of the power sets P(X) and P(Y ), respectively, obeying the axioms for a topological space.) Rather than identifying open sets via their inclusion in the topology—e.g., U ∈ TX —we will ...
... Suppose (X, TX ) and (Y, TY ) are topological spaces. (In other words, X and Y are sets, and TX and TY are subsets of the power sets P(X) and P(Y ), respectively, obeying the axioms for a topological space.) Rather than identifying open sets via their inclusion in the topology—e.g., U ∈ TX —we will ...
Algebraic Topology Lecture 1
... Compactness A subset W ⊆ X is a compact set if every open cover of W has a finite subcover. Connectedness A subset W ⊆ X is said to be connected if we can not find two disjoint open sets such that W is their union. Path Connectedness A subset W ⊆ X is said to be path connected if any two points in W ...
... Compactness A subset W ⊆ X is a compact set if every open cover of W has a finite subcover. Connectedness A subset W ⊆ X is said to be connected if we can not find two disjoint open sets such that W is their union. Path Connectedness A subset W ⊆ X is said to be path connected if any two points in W ...
PDF
... T = {A ⊆ X | X \ A is finite, or A = ∅}. In other words, the closed sets in the cofinite topology are X and the finite subsets of X. Analogously, the cocountable topology on X is defined to be the topology in which the closed sets are X and the countable subsets of X. The cofinite topology on X is t ...
... T = {A ⊆ X | X \ A is finite, or A = ∅}. In other words, the closed sets in the cofinite topology are X and the finite subsets of X. Analogously, the cocountable topology on X is defined to be the topology in which the closed sets are X and the countable subsets of X. The cofinite topology on X is t ...
Metric spaces
... be an open cover without finite sub covers. Call a set bad if no finite sub collection of covers it. Thus we assumed that itself is bad. Notice another property of bad set: if a finite number of other sets covers a bad set, one of them should be bad. Since there is a finite -net, one can find some b ...
... be an open cover without finite sub covers. Call a set bad if no finite sub collection of covers it. Thus we assumed that itself is bad. Notice another property of bad set: if a finite number of other sets covers a bad set, one of them should be bad. Since there is a finite -net, one can find some b ...
Lecture 5 Notes
... collection B of open sets containing x so that each neighborhood of x contains at least one of the elements of B. If X has a countable basis at each point x 2 X, we call X firstcountable. Definition 1.3. A topological space X is called second-countable if there exists a countable basis for its topol ...
... collection B of open sets containing x so that each neighborhood of x contains at least one of the elements of B. If X has a countable basis at each point x 2 X, we call X firstcountable. Definition 1.3. A topological space X is called second-countable if there exists a countable basis for its topol ...
