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Ascoli's Theorem Sign Video

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Definition

Let X be a space and let (Y, d) be a metric space. Give C(X, Y) the topology of compact convergence; let F be a subset of C(X, Y). a) If F is equicontinuous under d and the set F_d = {f(a)|f in F} has compact closure for each a in X, then F is contained in a compact subspace of C(C, Y). b) The converse holds if X is locally compact Hausdorff.

Source: Topology (second edition) by James R. Munkres

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> Ascoli's Theorem

+ Tychonoff Theorem *

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+ Discrete Topology *

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+ Standard Topology *

+ Lower Limit Topology *

+ Product Topology *

+ Homeomorphism *

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+ Hausdorff Space *

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+ Subspace Topology *

+ The Pasting Lemma *

+ Product Space *

+ Unit Circle *

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+ Bounded (Set) *

+ Metric Topology *

+ First Countability Axiom *

+ Standard Bounded Metric *

+ The Sequence Lemma *

+ Total Boundedness *

+ Uniform Limit Theorem *

+ Uniform Metric *

+ Uniform Topology *

+ Unit 2 Sphere *

+ Quoitent Topology *

+ Quotient Map *

+ Totally Disconnected *

+ Linear Continuum *

+ Punctured Euclidean Space *

+ Unit Sphere *

+ Local Connectedness *

+ Local Path Connectedness *

+ Path Connected *

+ Path Components *

+ Topological Group *

+ Limit Point Compactness *

+ Fininite Intersection Property *

+ Isolated Point *

+ Lebesgue Number *

+ Lebesgue Number Lemma *

+ Sequentially Compact *

+ The Tube Lemma *

+ Uniform Continuity Theorem *

+ Second Countable *

+ Completely Normal *

+ Directed Set *

+ First Countable *

+ Lindelof Space *

+ One Point Compactification *

+ Second Countability Axiom *

+ Sorgenfrey Plane *

+ Urysohn Lemma *

+ Completely Regular *

+ Baire Category Theorem *

+ Imbedding Theorem *

+ Locally Finite *

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+ Countably Locally Finite *

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+ Nagata Smirnov Metrization Thm *

+ Paracompact *

+ Smirnov Metrization Theorem *

+ Tietze Extension T Heorem *

+ Cauchy Sequence *

+ Equicontinuity *

+ Peano Space *

+ Point Open Topology *

+ Pointwise Boundedness *

+ Topologically Complete *

+ Baire Space *

+ Compact Open Topology *

+ Compactly Generated *

+ Empty Interior *

+ Evaluation Map *

+ General Position *

+ Topology Of Compact Convergence *

+ Affinely Independent *

+ Geometrically Independent *

+ Locally M Euclidean *

+ Topological Dimension *

+ Unit Ball (B^N) *

+ Uniform Convergence *

+ Connectedness *

- Ascoli's Theorem *

+ Box Topology *

+ Compactification *

+ Intermediate Value Theorem

+ Hairy Ball Theorem *

+ Local Compactness *

+ Partition Of Unity *

+ Quoitent Space *

+ Uniform Continuity *

+ Diffeomorphism *

+ Cauchy Integral Formula *

+ Fundamental Thm Of Algebra *

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