Category:Pages requiring proofs

P

• ► Pages requiring proofs of grade: A‎ (30 P)
• ► Pages requiring proofs of grade: A*‎ (8 P)
• ► Pages requiring proofs of grade: A**‎ (2 P)
• ► Pages requiring proofs of grade: B‎ (14 P)

P cont.

• ► Pages requiring proofs of grade: C‎ (32 P)
• ► Pages requiring proofs of grade: D‎ (14 P)
• ► Pages requiring proofs of grade: E‎ (3 P)
• ► Pages requiring proofs of grade: F‎ (1 P)

P cont.

• ► Pages requiring proofs of unknown grade‎ (25 P)
• ► Pages requiring proofs: Easy proofs‎ (39 P)

A

• Example:A bijective and continuous map that is not a homeomorphism
• A cap (B-C) = (A cap B) - C
• A compact and convex subset of Euclidean n-space with non-empty interior is a closed n-cell and its interior is an open n-cell
• A continuous map induces a homomorphism between fundamental groups
• A linear map is injective if and only if the image of every non-zero vector is a non-zero vector
• A map is continuous if and only if each point in the domain has an open neighbourhood for which the restriction of the map is continuous on
• A map is continuous if and only if the pre-image of every closed set is closed
• A pair of identical elements is a singleton
• A pre-measure on a semi-ring may be extended uniquely to a pre-measure on a ring
• A proper vector subspace of a topological vector space has no interior
• A sequence consisting of the nth terms of the sequences in a Cauchy sequence of elements in any little-L space is itself a Cauchy sequence of complex numbers
• A set is open if and only if every point in the set has an open neighbourhood contained within the set
• A subset of a topological space is disconnected if and only if it can be covered by two non-empty-in-the-subset and disjoint-in-the-subset sets that are open in the space itself
• A topological space is connected if and only if the only sets that are both open and closed in the space are the entire space itself and the emptyset
• A topological space is disconnected if and only if it is homeomorphic to a disjoint union of two or more non-empty topological spaces
• A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements
• A-(A-B) = A cap B
• Alec's c.d.f-sidestepping sampling method
• Axiom of completeness

B

• Basis for the tensor product

C

• Canonical injection of the subspace topology
• Canonical injections of the disjoint union topology
• Cantor's construction of the real numbers
• Task:Characteristic property of the coproduct topology
• Characteristic property of the direct product module
• Characteristic property of the direct sum module
• Characteristic property of the disjoint union topology
• Characteristic property of the product topology
• Characteristic property of the quotient topology
• Characteristic property of the subspace topology
• Characteristic property of the tensor product
• Closed map
• Comparison test for real series
• Concatenation of paths and loops (homotopy)

D

• Demonstrating why category arrows are best thought of as arrows and not functions
• Dense
• Direct product module
• Disjoint
• Disjoint union topology
• Distance from a point to a set
• Distributivity of intersections across unions

E

• Epsilon form of inequalities
• Equivalence classes are either equal or disjoint

E cont.

• Equivalence relation induced by a function
• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere/1 implies 2
• Equivalent conditions to a map being a quotient map
• Equivalent conditions to a set being bounded
• Equivalent conditions to a set being bounded/1 implies 2
• Equivalent conditions to a set being bounded/2 implies 1
• Equivalent conditions to a set being saturated with respect to a function
• Equivalent formulas
• Equivalent statements to a set being dense
• Every bijection yields an inverse function
• Every continuous map from a non-empty connected space to a discrete space is constant
• Every convergent sequence is Cauchy
• Every surjective map gives rise to an equivalence relation
• Exists functor

F

• Factoring a continuous map through the projection of an equivalence relation induced by that map yields an injective continuous map
• Factoring a function through the projection of an equivalence relation induced by that function yields an injection
• For a vector subspace of a topological vector space if there exists a non-empty open set contained in the subspace then the spaces are equal
• For any vector subspace of a Hilbert space the orthogonal complement and the closure of that subspace form a direct sum of the entire space

G

• Geometric distribution
• Given a Hilbert space and a non-empty, closed and convex subset then for each point in the space there is a closest point in the subset
• Given a homeomorphism all subspaces of the domain are homeomorphic to their image under the homeomorphism itself
• Given a topological manifold of dimension 2 or more and points p1, p2 and q where q is neither p1 nor p2 then a path from p1 to p2 is path-homotopic to a path that doesn't go through q
• Graph (topological manifold)

H

• Hausdorff space
• Hereditary system generated by
• Homeomorphic topological spaces have isomorphic fundamental groups
• Homotopy concatenation
• Homotopy invariance of loop concatenation
• Homotopy invariance of path concatenation
• Homotopy is an equivalence relation on the set of all continuous maps between spaces

I

• If A is a logical consequence of Gamma then the formula set of Gamma union the negation of A is not satisfiable
• If a surjective continuous map is factored through the canonical projection of the equivalence relation induced by that map then the yielded map is a continuous bijection
• If an inner product is non-zero then both arguments are non-zero
• If two charts are smoothly compatible with an atlas then they are smothly compatible with each other
• Infimum
• Integral of a positive function (measure theory)

L

• Lebesgue number lemma

M

• Mdm of a discrete distribution lemma
• Measure
• Module homomorphism
• Monotone convergence theorem for non-negative numerical measurable functions
• Monotonicity of the integral of non-negative extended-real-valued measurable functions with respect to a measure

O

• Operations on convergent sequences of real numbers

P

• Passing to the quotient (function)
• Pasting lemma
• Poisson race distribution
• Polynomial u-ring
• Pre-image sigma-algebra
• Pre-image sigma-algebra/Proof of claim: it is a sigma-algebra
• Proof that the fundamental group is actually a group
• Proof that the fundamental group is actually a group/Proof
• Properties of classes of sets closed under set-subtraction
• Properties of the pre-image of a function

Q

• Quotient module

R

• Template:Requires proof
• R^n is a topological vector space

S

• Semi-ring of half-closed-half-open intervals
• Set subtraction
• Sigma-algebra
• Example:Smooth map between the real line and the circle (considered as smooth manifolds)
• Sphere (topological manifold)
• Square lemma (of homotopic paths)
• Square root (real function)
• Subspace topology
• Symmetric difference

T

• Tensor product of tensors
• The basis criterion (topology)
• The canonical injections of the disjoint union topology are topological embeddings
• The closure of a linear subspace of a normed space is a linear subspace
• The composition of continuous maps is continuous
• The composition of end-point-preserving-homotopic paths with a continuous map yields end-point-preserving-homotopic paths
• The fundamental group
• The image of a compact set is compact
• The image of a connected set is connected
• The intersection of two sets is non-empty if and only if there exists a point in one set that is in the other set
• The little-L(C) spaces are complete
• The relation of path-homotopy is preserved under composition with continuous maps
• The ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions
• The vector space of all linear maps between two spaces
• Topology generated by a basis
• Trace sigma-algebra
• Trace sigma-algebra/Proof of claim that it actually is a sigma-algebra
• Trivial group

U

• Unique lifting property
• Urysohn's lemma