Review of metric spaces
... A Cauchy sequence in a metric space X is a sequence x1 , x2 , . . . with the property that for every ε > 0 there is N sufficiently large such that for i, j ≥ N we have d(xi , xj ) < ε. A point x ∈ X is a limit of that Cauchy sequence if for every ε > 0 there is N sufficiently large such that for i ≥ ...
... A Cauchy sequence in a metric space X is a sequence x1 , x2 , . . . with the property that for every ε > 0 there is N sufficiently large such that for i, j ≥ N we have d(xi , xj ) < ε. A point x ∈ X is a limit of that Cauchy sequence if for every ε > 0 there is N sufficiently large such that for i ≥ ...
Math F651: Homework 8 Due: March 29, 2017 Several of the
... Show that the homeomorphism group of a connected manifold acts transitively. In other words, show that if M is a connected manifold, then for any two points p and q in M there is a homeomorphism ψ ∶ M → M such that ψ(p) = q. Solution: Let G be the homeomorphism group of M. For each x ∈ M, let O x = ...
... Show that the homeomorphism group of a connected manifold acts transitively. In other words, show that if M is a connected manifold, then for any two points p and q in M there is a homeomorphism ψ ∶ M → M such that ψ(p) = q. Solution: Let G be the homeomorphism group of M. For each x ∈ M, let O x = ...
“The Lindelöf Property” - University of Birmingham
... The Lindelöf degree or number, L(X), of a space X is the smallest infinite cardinal κ for which every open cover has a subcover of cardinality at most κ. The hereditary Lindelöf degree, hL(X), of X is the supremum of the cardinals L(Y ) ranging over subspaces Y of X. The Lindelöf degree of a spa ...
... The Lindelöf degree or number, L(X), of a space X is the smallest infinite cardinal κ for which every open cover has a subcover of cardinality at most κ. The hereditary Lindelöf degree, hL(X), of X is the supremum of the cardinals L(Y ) ranging over subspaces Y of X. The Lindelöf degree of a spa ...
Baire Spaces and the Wijsman Topology
... For each nonempty open subset U in (F(X),Tbpd), i(U) has nonempty interior in (F(X),Twd). A mapping f: X → Y is called feebly open if f(U) has nonempty interior for every nonempty open set U of X. Similarly, a mapping f: X → Y is called feebly continuous whenever the pre-image of an open set ...
... For each nonempty open subset U in (F(X),Tbpd), i(U) has nonempty interior in (F(X),Twd). A mapping f: X → Y is called feebly open if f(U) has nonempty interior for every nonempty open set U of X. Similarly, a mapping f: X → Y is called feebly continuous whenever the pre-image of an open set ...
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.