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If and Only If
Last updated Nov 1, 2022
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A Compact Set that is not Closed
A Function into the the Product Sigma Algebra is Measureable iff its components are Measureable
A Function is Continuous iff it preserves Net Convergence
A Function is Continuous iff it takes Closed Sets back to Closed Sets
A Function is Measureable iff Preimages of Generating Sets are in the Source Sigma-Algebra
A Linear Equation System is Homogenous iff its constants are 0
A Measureable Function is less than or equal to another iff all its integrals are less than or equal to the others
A Nonempty Set is Compact in the Order Topology iff it is Tightly Bounded and Complete
A Normed Vector Space is Complete iff all Absolutely Convergent Series Converge
A Sequence Converges on the Extended Reals iff the number of Complete Upcrossings is Finite
A Set is Affine iff it contains all of its Affine Combinations
A Set is Closed iff it contains all Net Limits
A Set is Closed iff it is its own Closure
A Set is Closed in a First Countable Space iff it contains all its Sequential Limits
A Set is Closed in a Metric Space iff it contains all its Sequential Limits
A Set is Convex iff it contains all of its Convex Combinations
A Set is Open in the Metric Topology iff it contains a Ball around each Point
A Set is a Basis iff it is a Maximal Linearly Independent Set
A Set is a Basis iff it is a Minimal Spanning Set
A Submartingale has uniformly Bounded First Moment iff it has uniformly Bounded Positive Moment
A Subset of a Vector Space is a Subspace iff it contains all its Linear Combinations
A Subset of a Vector Space is a Subspace iff it is closed under scaling and addition
A Vector Space is Finite-Dimensional iff it has a Finite Spanning Set
A Vector Space is Infinite-Dimensional iff its Linearly Independent Set Size is Unbounded
A Vector Space is Infinite-Dimensional iff there exists an Infinite Linearly Independent Set
A finite measure is absolutely continuous wrt another iff small sets on the other measure are also small on it
A point in a First Countable Space is a Limit Point iff it is a Sequential Limit
A point is a Limit Point iff it is a Net Limit
A set is Closed iff it contains all its Limit Points
Affine Set
Augmenting Path Theorem of Matchings
Bounded Closed Intervals are Compact in the Order Topology iff the Order is Complete
Closure of Open Ball is a subset of Closed Ball
Compactness is a Topological Invariant
Connectedness is a Topological Invariant
Convergence of a Bounded Sequence is determined by Convergent Subsequences
Convex Set
Correspondence of Surjective Functions, Partitions, and Equivalence Relations
Distance Function
Extended Distance Function
Extended Norm
Heine Criterion
Holder Inequality
Inner Product
Kolmogorov Strong Law of Large Numbers
Kolmogorov Three Series Theorem
Levy's Theorem
Liminf of Set Sequence
Limsup of Set Sequence
Linearly Independent Set Size is Unbounded iff there exists an Infinite Linearly Independent Set
Local Path-Connectedness is a Topological Invariant
Locally Path-Connected Spaces are Path-Connected iff they are Connected
Log Concavity Inequality
Martingale Convergence Theorem
Norm
Monotone Nets Converge to their Supremum
Partial Ordering
Path-Connectedness is a Topological Invariant
Sequence Convergence
Sum of Independent Random Variables is in L1 if and only if Random Variables are in L1
The Induced Sigma Algebra of a Borel Function is Finite iff it is Simple
Topological Manifolds Connected iff Path-Connected
Components Converge iff Convergence in Product Metric Space
Upcrossing Inequality
Vertex Cover
Interactive Graph