Topology Flashcards
Define a topological space
A topological space (X, T~) is a non-empty set X together with a family T~ of subsets of X satisfying: T1: X, ∅ ∈ T~, T2: U, V ∈ T ⇒ U∩V ∈ T~, and T3: Ui ∈ T~ for all i ∈ I ⇒ the union of Ui ∈ T~.
Define an open set
If (X, T~) is a topological space, then the sets in T~ are called the open sets of X.
Define the discrete topological space
Let X be any non-empty set. Then, the discrete topology on X is the set of all subsets of X.
Define the indiscrete topological space
Let X be any non-empty set. Then, the indiscrete topology on X is {X, ∅}.
Define a co-finite topology
Let X be any non-empty set. The co-finite topology on X consists of the empty set together with every subset U of X such that X\U is finite.
Define a metrizable topological space
A topological space (X, T~) metrizable if it arises from a metric space (X, d), where T~ is defined to be the set of open sets in (X, d).
Define topologically equivalent metrics
Two metrics on a set are topologically equivalent if they give rise to the same topology.
Define a coarser topology
Given two topologies T~1, T~2 on the same set, we say T~1 is coarser than T~2 if T~1 ⊆ T~2.
Define a closed set
Let (X, T~) be a topological space. A subset V of X is closed in X if X\V is open in X.
Define a convergent sequence
A sequence (xn) in a topological space X converges to a point x ∈ X if given any open set U containing x there exists an integer N such that xn ∈ U for all n > N.
Define a continuous function
Suppose that (X, T~X) and (Y, T~Y) are topological spaces and that f: X → Y is a map. f is continuous if U ∈ T~Y ⇒ f^(−1)U ∈ T~X.
Define a homeomorphism
A homeomorphism between topological spaces X and Y is a bijection f: X → Y such that f and f^(−1) are continuous.
Define homeomorphic topological spaces
Two topological spaces X and Y are homeomorphic if there exists a homeomorphism between them.
Define the closure of a set
The closure of a set A is the intersection of all closed sets containing A.
Define a dense set
A subset A in X is dense if A = X.
Define an accumulation point
A point x ∈ X such that for any open U ⊆ X with x ∈ U, (U{x})∩A ≠ ∅ is called an accumulation point of A.
Define the interior of a set
The interior of a set A is the union of all open sets contained within A.
Define the boundary of a set
The boundary of a set A is given by ∂A = closure(A)\interior(A).
Give the first separation axiom
A topological space satisfies the first separation axiom if for any two distinct points a, b in X, there exists an open set U containing a and not b.
Define Hausdorff
A topological space X is said to be Hausdorff if given any two distinct points x, y in X, there exist disjoint open sets U, V with x ∈ U, y ∈ V.
Define a subspace
Let (X, T~ ) be a topological space and let A be a non-empty subset of X. The subspace or induced topology on A is T~A = {A ∩ U : U ∈ T~}
Define a topological basis
Given a topological space (X, T~), a collection of subsets of X is a basis for T if B ⊆ T~ and every set in T~ can be expressed as a union of sets in B.
Define the product topology
Let (X, T~X), (Y, T~Y) be topological spaces. Then, the family of all unions of the basis B~(X×Y) = {U x V : U ∈ T~X, V ∈ T~Y} gives the topology T~(XxY) for XxY.
Define a disjoint union
Let X and Y be sets. Their disjoint union is (X x {0}) ∪ (Y x {1}).
Define a cover
A family {Ui: i ∈ I} of subsets of a space X is called a cover if X = infinite union of Ui.
Define an open cover
A cover where every set in the covering family is open.
Define a subcover
A subcover of a cover {Ui: i ∈ I} for a space X is a subfamily {Uj: j ∈ J} for some subset J ⊆ I such that {Uj: j ∈ J} is still a cover for X.
Define compact
A topological space is compact if every cover has a finite subcover.
Define a compact subset
A subset A of a topological space X is compact if it is compact when endowed with the subspace topology.
State the extreme value theorem
If f: X → R is a continuous real-valued function on a compact space X then f is bounded and attains its bounds.
State the generalised Heine-Borel theorem
Any closed bounded subset of (R^n, ||.||_i), where i ∈ {1, 2, ∞}, is compact.
Define sequentially compact
A topological space X is sequentially compact if every sequence in X has a subsequence converging to a point in X.
State Bolzano Weierstrass’ Theorem
In a compact topological space X every infinite subset has accumulation points.
Define a net
Given a real number ε > 0, an ε-net for a metric space X is a subset N ⊆ X such that {B(x, ε) : x ∈ N} is a cover of X.
Define a relation
Let X be a set. A relation on X is a subset R of XxX.
Define an equivalence relation
An equivalence relation R on X is a relation that is reflexive, symmetric and transitive.
Define an equivalence class
Given an equivalence relation R, the equivalence class of an element x ∈ X is the set [x] = {y ∈ X : yRx}.
Define a quotient space
Given an equivalence relation R, the quotient space of X with respect to R, X/R is the set of equivalence classes of R.
Define a partition
A partition of a non-empty set X is a collection {Ui : i ∈ I} of subsets of X such that Ui ≠ ∅ for every i ∈ I, Ui ∩ Uj = ∅ for every i ≠ j and the union of all Ui = X.
Define a collapsing map
Let p: X → X/R be the map that assigns to each point x in X the equivalence class [x]. Then p is the collapsing map for X with respect to R.
Define the quotient topology
Let (X, T~) be a topological space, and R an equivalence relation on X. Then, the family T’~ of subsets U’ in X/R such that p^(−1)(U’) is the quotient topology for X/R.
Define identified points
Given two x, y in a topological space X and an equivalence relation R, we say that the points are identified if xRy.
Define a torus
A square with both pairs of parallel edges identified in the same orientation and the four corners are identified.
Define a transitive closure
Let X be a set and R be a relation on X that is reflexive and symmetric, but not necessarily transitive. Then, the transitive closure of R is the relation R* given by xR*x’ ⇔ there is a finite sequence of points x1,… , xn in X such that x = x1Rx2R . . . Rxn−1Rxn = x’.
Define the Klein bottle
A square with one pair of parallel edges identified in the same orientation and one pair identified in the reverse orientation and the four corners are identified.
Define a saturated set
Let X be a topological space, and let R be an equivalence relation on X. Then a subset A of X is saturated with respect to R if it is a union of equivalence classes.
Define the Mobius band
A square with one pair of parallel edges identified in the opposite orientation.
Define the real n-dimensional projective space
The real n-dimensional projective space RP^n is the quotient space of R^(n+1){0} with respect to the equivalence relation R where xRy if and only if there exists λ ≠ 0 such that x = λy.
Define an open mapping
A function f: X → Y between topological spaces is an open mapping if, for each open set U in X , f(U) is open in Y.
Define a quotient map
A map p: X → Y between topological spaces is a quotient map if p is a surjective continuous open mapping.
Define an n-simplex
The standard n-simplex is the set ∆n = {(x1,… , xn+1) ∈ R^(n+1): xi ≥ 0 ∀i and the sum of xi = 1}.
Define simplex vertices
The vertices of ∆n, denoted V(∆n), are those points (x1, …, xn+1) in ∆n where xi = 1 for some i (and hence xj = 0 for all j ≠ i).
Define simplex faces
For each non-empty subset A of {1, …, n+1} there is a corresponding face of ∆n, which is {(x1,… , xn+1) ∈ ∆n: xi = 0 ∀i ∉ A}.
Define a simplex’s inside
The inside of ∆n is inside(∆n) = {(x1, …, xn+1) ∈ ∆n : xi > 0 ∀i}.
Define face inclusion
A face inclusion of a standard m-simplex ∆m into a standard n-simplex ∆n (where m < n) is a function ∆m → ∆n that is the restriction of an injective linear map R^(m+1) → R^(n+1) which sends the vertices of ∆m to vertices of ∆n.
Define an abstract simplicial complex
An abstract simplicial complex is a pair (V, Σ), where V is a set and Σ is a set of non-empty finite subsets of V such that:
1. For each v ∈ V, the 1-element set {v} is in Σ.
2. If σ is an element of Σ, so is any non-empty subset of σ.
Define a finite abstract simplicial complex
A simplicial complex where the set of vertices is finite.
Define a topological realisation
Given an abstract simplicial complex K = (V, Σ), the topological realisation |K| of is the space obtained by the following procedure: For each σ ∈ Σ, take a copy of the standard n-simplex, where n+1 is the number of elements of σ. Denote this simplex by ∆σ. Label its vertices with the elements of σ. Then, whenever σ ⊂ τ ∈ Σ, identify ∆σ with a subset of ∆τ, via the face inclusion which sends the elements of σ to the corresponding elements of τ.
Define a triangulation
A triangulation of a space X is a simplicial complex K together with a choice of homeomorphism |K| → X.
Define a simplicial circle
Let K be a simplicial complex with vertices {v1, …, vn} for some n ≥ 3, and having just 0-simplices and 1-simplices, where the 1-simplices are precisely {vi , vi+1} for each i between 1 and n−1, and {vn, v1}. Then K is a simplicial circle.
Define a subcomplex
A subcomplex of a simplicial complex K = (V, Σ) is a simplicial complex K’ = (V’, Σ’), where V’ ⊆ V and Σ’ ⊆ Σ.
Define the span of a subcomplex
For a simplicial complex K = (V, Σ) and a subset V’ of the vertex set V, the subcomplex spanned by V’ has vertex set V’ and consists of all simplices in Σ that have all their vertices in V’.
Define the link of a vertex
Let K = (V, Σ) be a simplicial complex, and let v ∈ V be a vertex. Then, the link of v is the subcomplex with vertex set {w ∈ V{v}: {v, w} ∈ Σ} and with simplices σ such that v ∉ σ and σ ∪ {v} ∈ Σ.
Define the star of a vertex
Let K = (V, Σ) be a simplicial complex, and let v ∈ V be a vertex. Then, the star of v is the union of {inside(σ) : σ ∈ Σ and v ∈ σ}.
Define an edge path
An edge path in a simplicial complex K is a sequence of vertices (v0, …, vn) such that for every i, {vi , vi+1} is a simplex of K.
Define an n-manifold
An n-manifold is a Hausdorff topological space M such that every point of M lies in an open set that is homeomorphic to an open set in R^n.
Define a surface
A surface is a 2-manifold.
Define polygons with a complete set of side identifications
Let P be a finite-sided convex polygon in R^2, with an even number of sides. Arrange these sides into pairs and for each pair, identify the two sides. More specifically, suppose that e and e’ are two sides of a pair. Let e run from (x0, y0) to (x1, y1), and let e’ run from (x’0, y’0) to (x’1, y’1). Then, as t runs from 0 to 1, the point (1−t)(x0, y0) + t(x1, y1) lies on e and (1−t)(x’0 , y’0) + t(x’1, y’1) lies on e’. For each t ∈ [0, 1], we identify these two points (possibly reversing the direction of the identification. This construction is a polygon with a complete set of side identifications.
Define a word
An ordered string of letters, possibly with ( )^-1 signs, where we start at a vertex v0 and go around the polygon until we reach v0 again, with each letter representing an identification and ()^-1 denoting the backwards identification.
Define a g-holed torus
Let M0 be represented by the word xx^(−1)yy^(−1). For g ≥ 1, let Mg be the surface obtained from the word x1y1x1^(−1)y1^(-1)1…xgygxg^(-1)yg^(−1).
Define a surface with h crosscaps
Let N1 be represented by the word xxyy−1. For h ≥ 2, let Nh be the surface obtained from the word x1x1x2x2 . . . xhxh.
Define adding a handle
Let S be a surface and T be a torus. Pick subsets D1 and D2 of S and T, each of which is homeomorphic to a closed disc. Remove the interiors of D1 and D2 from S and T. Let S’ and T’ be the resulting spaces. What remains of each disc is C1 and C2. Now, pick a homeomorphism φ: C1 → C2 and form the following topological space: start with the disjoint union of S’ and T’ and form the quotient space, where each point x on C1 is identified with φ(x) on C2, but no other points are identified. This topological space is said to be obtained from S by adding a handle.
Define adding a crosscap
Let S be a surface and T be RP^2. Pick subsets D1 and D2 of S and T, each of which is homeomorphic to a closed disc. Remove the interiors of D1 and D2 from S and T. Let S’ and T’ be the resulting spaces. What remains of each disc is C1 and C2. Now, pick a homeomorphism φ: C1 → C2 and form the following topological space: start with the disjoint union of S’ and T’ and form the quotient space, where each point x on C1 is identified with φ(x) on C2, but no other points are identified. This topological space is said to be obtained from S by adding a crosscap.
Define a closed combinatorial surface
A closed combinatorial surface is a connected finite simplicial complex K such that for every vertex v of K, the link of v is a simplicial circle.
Give the classification theorem
Every closed combinatorial surface is homeomorphic to one of the manifolds Mg, for some g ≥ 0, or Nh, for some h ≥ 1.
