Quotient map & adjuction space

Suppose f : X→Y is a quotient map between spaces such that it maps a subset A→X injectively into Y and maps X\A onto Y\p(A). Then Y is homeomorphic to the adjuction space X∪_p|X\A(Y\p(A)).

pf> Let A'=X\A, B=p(A), B'=Y\p(A). Let π:X∪B'→X∪_p|A'B' be the quotient map defining the adjunction space, and let φ:X∪_p|A'B'→Y be a map defined in the obvious way. Then clearly φ is bijective. I'll show it's homeomorphism.
Suppose U⊆Y is open. π^-1φ^-1(U)=p^-1(U) so is open in X, and π^-1φ^-1(U)∩B' =U∩B' is obviously open in B' so π^-1φ^-1(U) is open in X∪B', and hence φ^-1(U) is open in X∪_p|A'B'.
Now suppose V⊆Y is a subset such that φ^-1(V) is open. Then π^-1φ^-1(V) is open. Since p^-1(V)=π^-1φ^-1(V), V is open in Y. QED

Primes and primaries in module

M an R-module.
An element r∈R is a zero divisor in M if rm = 0 for some m≠0. r is a zero in M if rm = 0 ∀m∈M. r is a nilpotent in M if some positive power of r is a zero in M. Provided M≠0, zeros ⊆ nilpotents ⊆ zero divisors.
Let N⊆M be a submodule. N is a prime submodule if N≠M and every zero divisor in M/N is a zero in M/N. N is a primary submodule if N≠M and every nilpotent in M/N is a zero divisor in M/N. Then prime submodules are primary.
Let P⊆M be a prime submodule. Then (P:M) is a prime ideal.
Let Q⊆M be a primary submodule. Then (Q:M) is a primary ideal.
If one defines rad(N)=rad((N:M)) for submodules N⊆M, then rad(Q)=rad((Q:M)).

Continous map from topological sum

Let {X_α}_α, Y be disjoint spaces, and f_α:X_α→Y continuous maps. Then the natural induced map from the topological sum ∪_αX_α to Y is continuous.

Adjunction space and quotient topology

Let X and Y be disjoint spaces, and A⊆X a subset.
Let f: A→Y be a function. Let Z=(X-A)∪Y be the disjoint union of sets. Then there is a natural surjection from X∪Y(as topological sum) to Z. The quotient topology on Z induced by this function is none other than the adjuction space X∪_fY.

Adjunction space

Suppose a function between spaces f: X→Y.
The adjuction space X∪_fY is homeomorphic to Y if and only if f is continuous.

Coherent topology

Suppose X is a space coherent with a family of its subspaces {X_α}_α.
Then a map f: X →Y is continuous if and only if f_|Xα:X_α→Y is continuous for every α.

Topological sum(Munkres)

X a space, {U_α}_α a family of disjoint open sets of X. X is a topological sum of U_α's if X is a disjoint union of them.