Topology

Question 1

Define a topological space

Answer 1

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~.

Question 2

Define an open set

Answer 2

If (X, T~) is a topological space, then the sets in T~ are called the open sets of X.

Question 3

Define the discrete topological space

Answer 3

Let X be any non-empty set. Then, the discrete topology on X is the set of all subsets of X.

Question 4

Define the indiscrete topological space

Answer 4

Let X be any non-empty set. Then, the indiscrete topology on X is {X, ∅}.

Question 5

Define a co-finite topology

Answer 5

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.

Question 6

Define a metrizable topological space

Answer 6

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).

Question 7

Define topologically equivalent metrics

Answer 7

Two metrics on a set are topologically equivalent if they give rise to the same topology.

Question 8

Define a coarser topology

Answer 8

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.

Question 9

Define a closed set

Answer 9

Let (X, T~) be a topological space. A subset V of X is closed in X if X\V is open in X.

Question 10

Define a convergent sequence

Answer 10

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.

Question 11

Define a continuous function

Answer 11

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.

Question 12

Define a homeomorphism

Answer 12

A homeomorphism between topological spaces X and Y is a bijection f: X → Y such that f and f^(−1) are continuous.

Question 13

Define homeomorphic topological spaces

Answer 13

Two topological spaces X and Y are homeomorphic if there exists a homeomorphism between them.

Question 14

Define the closure of a set

Answer 14

The closure of a set A is the intersection of all closed sets containing A.

Question 15

Define a dense set

Answer 15

A subset A in X is dense if A = X.

Question 16

Define an accumulation point

Answer 16

A point x ∈ X such that for any open U ⊆ X with x ∈ U, (U{x})∩A ≠ ∅ is called an accumulation point of A.

Question 17

Define the interior of a set

Answer 17

The interior of a set A is the union of all open sets contained within A.

Question 18

Define the boundary of a set

Answer 18

The boundary of a set A is given by ∂A = closure(A)\interior(A).

Question 19

Give the first separation axiom

Answer 19

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.

Question 20

Define Hausdorff

Answer 20

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.

Question 21

Define a subspace

Answer 21

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~}

Question 22

Define a topological basis

Answer 22

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.

Question 23

Define the product topology

Answer 23

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.

Question 24

Define a disjoint union

Answer 24

Let X and Y be sets. Their disjoint union is (X x {0}) ∪ (Y x {1}).

Question 25

Define a cover

Answer 25

A family {Ui: i ∈ I} of subsets of a space X is called a cover if X = infinite union of Ui.

Question 26

Define an open cover

Answer 26

A cover where every set in the covering family is open.

Question 27

Define a subcover

Answer 27

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.

Question 28

Define compact

Answer 28

A topological space is compact if every cover has a finite subcover.

Question 29

Define a compact subset

Answer 29

A subset A of a topological space X is compact if it is compact when endowed with the subspace topology.

Question 30

State the extreme value theorem

Answer 30

If f: X → R is a continuous real-valued function on a compact space X then f is bounded and attains its bounds.

Question 31

State the generalised Heine-Borel theorem

Answer 31

Any closed bounded subset of (R^n, ||.||_i), where i ∈ {1, 2, ∞}, is compact.

Question 32

Define sequentially compact

Answer 32

A topological space X is sequentially compact if every sequence in X has a subsequence converging to a point in X.

Question 33

State Bolzano Weierstrass' Theorem

Answer 33

In a compact topological space X every infinite subset has accumulation points.

Question 34

Define a net

Answer 34

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.

Question 35

Define a relation

Answer 35

Let X be a set. A relation on X is a subset R of XxX.

Question 36

Define an equivalence relation

Answer 36

An equivalence relation R on X is a relation that is reflexive, symmetric and transitive.

Question 37

Define an equivalence class

Answer 37

Given an equivalence relation R, the equivalence class of an element x ∈ X is the set [x] = {y ∈ X : yRx}.

Question 38

Define a quotient space

Answer 38

Given an equivalence relation R, the quotient space of X with respect to R, X/R is the set of equivalence classes of R.

Question 39

Define a partition

Answer 39

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.

Question 40

Define a collapsing map

Answer 40

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.

Question 41

Define the quotient topology

Answer 41

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.

Question 42

Define identified points

Answer 42

Given two x, y in a topological space X and an equivalence relation R, we say that the points are identified if xRy.

Question 43

Define a torus

Answer 43

A square with both pairs of parallel edges identified in the same orientation and the four corners are identified.

Question 44

Define a transitive closure

Answer 44

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’.

Question 45

Define the Klein bottle

Answer 45

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.

Question 46

Define a saturated set

Answer 46

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.

Question 47

Define the Mobius band

Answer 47

A square with one pair of parallel edges identified in the opposite orientation.

Question 48

Define the real n-dimensional projective space

Answer 48

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.

Question 49

Define an open mapping

Answer 49

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.

Question 50

Define a quotient map

Answer 50

A map p: X → Y between topological spaces is a quotient map if p is a surjective continuous open mapping.

Question 51

Define an n-simplex

Answer 51

The standard n-simplex is the set ∆n = {(x1,… , xn+1) ∈ R^(n+1): xi ≥ 0 ∀i and the sum of xi = 1}.

Question 52

Define simplex vertices

Answer 52

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).

Question 53

Define simplex faces

Answer 53

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}.

Question 54

Define a simplex's inside

Answer 54

The inside of ∆n is inside(∆n) = {(x1, …, xn+1) ∈ ∆n : xi > 0 ∀i}.

Question 55

Define face inclusion

Answer 55

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.

Question 56

Define an abstract simplicial complex

Answer 56

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.

Question 57

Define a finite abstract simplicial complex

Answer 57

A simplicial complex where the set of vertices is finite.

Question 58

Define a topological realisation

Answer 58

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.

Question 59

Define a triangulation

Answer 59

A triangulation of a space X is a simplicial complex K together with a choice of homeomorphism |K| → X.

Question 60

Define a simplicial circle

Answer 60

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.

Question 61

Define a subcomplex

Answer 61

A subcomplex of a simplicial complex K = (V, Σ) is a simplicial complex K’ = (V’, Σ’), where V’ ⊆ V and Σ’ ⊆ Σ.

Question 62

Define the span of a subcomplex

Answer 62

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’.

Question 63

Define the link of a vertex

Answer 63

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.

Question 64

Define the star of a vertex

Answer 64

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}.

Question 65

Define an edge path

Answer 65

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.

Question 66

Define an n-manifold

Answer 66

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.

Question 67

Define a surface

Answer 67

A surface is a 2-manifold.

Question 68

Define polygons with a complete set of side identifications

Answer 68

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.

Question 69

Define a word

Answer 69

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.

Question 70

Define a g-holed torus

Answer 70

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).

Question 71

Define a surface with h crosscaps

Answer 71

Let N1 be represented by the word xxyy−1. For h ≥ 2, let Nh be the surface obtained from the word x1x1x2x2 . . . xhxh.

Question 72

Define adding a handle

Answer 72

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.

Question 73

Define adding a crosscap

Answer 73

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.

Question 74

Define a closed combinatorial surface

Answer 74

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.

Question 75

Give the classification theorem

Answer 75

Every closed combinatorial surface is homeomorphic to one of the manifolds Mg, for some g ≥ 0, or Nh, for some h ≥ 1.