Category:Metric Space

1

• UTLOC:1

A

• A function is continuous if and only if the pre-image of every basis element is open
• A monotonically increasing sequence bounded above converges
• A set is bounded if and only if for all points in the space there is a positive real such that the distance from that point to any point in the set is less than the positive real
• A set is dense if and only if every non-empty open subset contains a point of it
• A subset of a topological space is open if and only if it is a neighbourhood to all of its points
• An open ball contains another open ball centred at each of its points
• An open set is a neighbourhood to all of its points

B

• Boundary (topology)
• Bounded
• Bounded (linear map)
• Bounded (sequence)
• Bounded (set)
• Bounded linear map
• Bounded sequence
• Bounded set

C

• Cauchy sequence
• Cauchy sequence/Definition
• Cauchy sequence/Short definition
• Closed interval
• Closed set
• Closed sets
• Closure (set, topology)
• Closure of a set in a topological space
• Compact
• Compact (topology)
• Compactness
• Comparison test for real series
• Comparison test for real series/Statement
• Complete metric space
• Completeness
• Connected topological space
• Constant loop
• Constant loop based at
• Continuity
• Continuous
• Continuous function
• Continuous map
• Continuous map/Claim: continuous iff continuous at every point
• Continuous maps
• Convergence (sequence)
• Convergence of a sequence
• Convergent (sequence)
• Convergent sequence
• Converges (sequence)
• Cover
• Covering

D

• Dense

D cont.

• Dense set
• Differentiation
• Discrete metric
• Discrete metric and topology
• Discrete metric and topology/Metric space definition
• Discrete metric and topology/Summary
• Discrete topology
• Distance from a point to a set

E

• Equivalence of Cauchy sequences
• Equivalence of Cauchy sequences/Definition
• Equivalence of Cauchy sequences/Proof
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere
• Equivalent conditions to a set being bounded
• Equivalent conditions to a set being bounded/Statement
• Equivalent statements to a set being dense
• Equivalent statements to compactness of a metric space
• Equivalent statements to compactness of a metric space/Statement
• Euclidean distance
• Euclidean metric
• Every convergent sequence is Cauchy
• Every lingering sequence has a convergent subsequence
• Every map from a space with the discrete topology is continuous
• Every sequence in a compact space is a lingering sequence

G

• Given two open balls sharing the same centre but with differing radius then the one defined to have a strictly smaller radius is contained in the other

H

• Hausdorff
• Homeomorphism

I

• If a set is a neighbourhood to all of its points then it is open
• If a subsequence of a Cauchy sequence converges then the Cauchy sequence itself also converges
• If the intersection of two open balls is non-empty then for every point in the intersection there is an open ball containing it in the intersection
• Inner-product space
• Interior
• Interior (topology)
• Interior point (topology)
• Isometry

L

• Lebesgue number
• Lebesgue number lemma
• Limit (sequence)
• Limit point
• Lingering sequence
• List of topological properties
• Loop
• Loop (topology)
• Loop concatenation
• Loops

M

• Metric
• Metric induced by a norm
• Metric induced by the norm
• Metric space

M cont.

• Metric spaces
• Template:Metric spaces navbox
• Metric subspace
• Metric topology
• Metrically bounded set
• Metrics

N

• Neighbourhood
• Neighbourhoods
• Norm
• Normed spaces
• Null sequence

O

• Open ball
• Open balls
• Open cover
• Open covering
• Open neighbourhoods
• Open set
• Open sets
• Operations on convergent sequences of real numbers

P

• Path (topology)

R

• Real line
• Real sequences

S

• Sequential compactness
• Series (summation)
• Set of continuous functions
• Subspace
• Subspace (metric space)

T

• The ell p spaces
• The interior of a set in a topological space is equal to the union of all interior points of that set
• The norm of a space is a uniformly continuous map with respect to the topology it induces
• The real line
• The reals
• The set of all open balls of a metric space are able to generate a topology and are a basis for that topology
• The set of continuous functions between topological spaces
• The stages of a homotopy are continuous
• TOP (category)
• Topological space induced by the metric space
• Topological space/Definition
• Topology
• Topology induced by a metric
• Topology induced by the metric
• Topology theorems
• Totally bounded
• Trivial

U

• Unbounded
• Uniform continuity
• Uniformly continuous
• Usual topology of the reals