Category:Metric Space Theorems
Pages in category "Metric Space Theorems"
The following 36 pages are in this category, out of 36 total.
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
Comparison test for real series
Comparison test for real series/Statement
Continuous map/Claim: continuous iff continuous at every point
Discrete metric and topology/Summary
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
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
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
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
Lebesgue number lemma
Operations on convergent sequences of real numbers
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 set of all open balls of a metric space are able to generate a topology and are a basis for that topology
The stages of a homotopy are continuous
Topology induced by a metric
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