GEOMETRY Review NOTES in word document
... etc. We write these as ft2 , m2, but still read them as square feet, square meters, etc. Example: Planes, Polygons (chart attached and on-line), circles, semicircles, etc. A plane is a 2-D object (flat surface) with infinite length and width. It is named by a cursive, uppercase letter or 3 NON-COLLI ...
... etc. We write these as ft2 , m2, but still read them as square feet, square meters, etc. Example: Planes, Polygons (chart attached and on-line), circles, semicircles, etc. A plane is a 2-D object (flat surface) with infinite length and width. It is named by a cursive, uppercase letter or 3 NON-COLLI ...
Pants decompositions of random surfaces
... construct it by gluing some hyperbolic pairs of pants with total boundary length ≤ L. A hyperbolic pair of pants is determined by its boundary lengths, so the number (really, Weil-Petersson volume) of possible surfaces with total pants length ≤ L is governed by the number of possible ways of choosin ...
... construct it by gluing some hyperbolic pairs of pants with total boundary length ≤ L. A hyperbolic pair of pants is determined by its boundary lengths, so the number (really, Weil-Petersson volume) of possible surfaces with total pants length ≤ L is governed by the number of possible ways of choosin ...
The Lattice of Domains of an Extremally Disconnected Space 1
... X. Recall that A is said to be a domain in X provided Int A ⊆ A ⊆ Int A (see [24], [11]). Recall also that A is said to be a(n) closed (open) domain in X if A = Int A (A = Int A, resp.) (see e.g. [14], [24]). It is well-known that for a given topological space all its closed domains form a Boolean l ...
... X. Recall that A is said to be a domain in X provided Int A ⊆ A ⊆ Int A (see [24], [11]). Recall also that A is said to be a(n) closed (open) domain in X if A = Int A (A = Int A, resp.) (see e.g. [14], [24]). It is well-known that for a given topological space all its closed domains form a Boolean l ...
FINITE TOPOLOGICAL SPACES 1. Introduction: finite spaces and
... Theorem 2.13. The homeomorphism classes of finite spaces are in bijective correspondence with M . The number of sets in a minimal basis for X determines the size of the corresponding matrix, and the trace of the matrix is the number of elements of X. Proof. We work with minimal bases for the topolog ...
... Theorem 2.13. The homeomorphism classes of finite spaces are in bijective correspondence with M . The number of sets in a minimal basis for X determines the size of the corresponding matrix, and the trace of the matrix is the number of elements of X. Proof. We work with minimal bases for the topolog ...
Topological Groups Part III, Spring 2008
... group for each α ∈ A. Their complete direct product is G = α Gα equipped with the usual product topology τG and with multiplication given by (x ×G y)α = xα ×α yα . Definition 4.2. Suppose A is non-empty and (Gα , ×α ) is a group Q for each α ∈ A. Their direct product is the subgroup of the group G = ...
... group for each α ∈ A. Their complete direct product is G = α Gα equipped with the usual product topology τG and with multiplication given by (x ×G y)α = xα ×α yα . Definition 4.2. Suppose A is non-empty and (Gα , ×α ) is a group Q for each α ∈ A. Their direct product is the subgroup of the group G = ...
On Quasi Compact Spaces and Some Functions Key
... 2. Every proper A-closed set is quasi compact with respect to X. Proof: (1) ⇒ (2): Let A be a proper A-closed subset of X. Let {Uα : α ∈ Λ} be a cover of A by cozero sets of X. Now for each x ∈ X − A, there is a cozero set Vx such that Vx ∩ A is finite. Since {Uα : α ∈ Λ} ∪ {Vx : x ∈ X − A} is a coz ...
... 2. Every proper A-closed set is quasi compact with respect to X. Proof: (1) ⇒ (2): Let A be a proper A-closed subset of X. Let {Uα : α ∈ Λ} be a cover of A by cozero sets of X. Now for each x ∈ X − A, there is a cozero set Vx such that Vx ∩ A is finite. Since {Uα : α ∈ Λ} ∪ {Vx : x ∈ X − A} is a coz ...
THE MEASURE OF ONE ANGLE IS 38 LESS THAN THE MEASURE
... SEGMENTS CALLED SIDES. 2) EACH SIDE INTERSECTS EXACTLY TWO SIDE, ONE AT EACH ENDPOINT, SO THAT NO TWO SIDES WITH A COMMON ENDPOINT ARE COLLINEAR EACH ENDPOINT OF A SIDE IS A VERTEX OF THE POLYGON. ...
... SEGMENTS CALLED SIDES. 2) EACH SIDE INTERSECTS EXACTLY TWO SIDE, ONE AT EACH ENDPOINT, SO THAT NO TWO SIDES WITH A COMMON ENDPOINT ARE COLLINEAR EACH ENDPOINT OF A SIDE IS A VERTEX OF THE POLYGON. ...
Geometry Review KCAS 7.G.3-6, 8.G.2, 8.G.4, 8.G.5, 8.G.9 W. 11
... etc. We write these as ft2 , m2, but still read them as square feet, square meters, etc. Example: Planes, Polygons (chart attached and on-line), circles, semicircles, etc. A plane is a 2-D object (flat surface) with infinite length and width. It is named by a cursive, uppercase letter or 3 NON-COLLI ...
... etc. We write these as ft2 , m2, but still read them as square feet, square meters, etc. Example: Planes, Polygons (chart attached and on-line), circles, semicircles, etc. A plane is a 2-D object (flat surface) with infinite length and width. It is named by a cursive, uppercase letter or 3 NON-COLLI ...
Chapter 3 Connected Topological Spaces
... It is to be noted that if X is a connected topological space and f : X → Y is a continuous function, where Y is any topological space, then the image f (X) is also a connected topological space. Here we will have to consider f (X) as a subspace of the given topological space Y . Also if f : X → Y is ...
... It is to be noted that if X is a connected topological space and f : X → Y is a continuous function, where Y is any topological space, then the image f (X) is also a connected topological space. Here we will have to consider f (X) as a subspace of the given topological space Y . Also if f : X → Y is ...
two classes of locally compact sober spaces
... A space is called stably compact if it is compact, locally compact, coherent, and sober. One of the principal results of the theory of stably compact spaces [2, page 474] is that a T0 space is stably compact if and only if the patch topology is a compact Hausdorff topology. The patch topology is an i ...
... A space is called stably compact if it is compact, locally compact, coherent, and sober. One of the principal results of the theory of stably compact spaces [2, page 474] is that a T0 space is stably compact if and only if the patch topology is a compact Hausdorff topology. The patch topology is an i ...
COMPACTIFICATIONS WITH DISCRETE REMAINDERS all
... (B) and (C) are equivalent when aX - X is locally compact by Theorem 7.2 of XI, [5]. That (C) and (D) are equivalent is obvious, as is (A) implies (E). Now assume (E) and let K be a compact set of countable character containing R(X). Let {Gn\n £ N] be a countable open neighborhood base for K and tak ...
... (B) and (C) are equivalent when aX - X is locally compact by Theorem 7.2 of XI, [5]. That (C) and (D) are equivalent is obvious, as is (A) implies (E). Now assume (E) and let K be a compact set of countable character containing R(X). Let {Gn\n £ N] be a countable open neighborhood base for K and tak ...
Generically there is but one self homeomorphism of the Cantor set
... ˙ is a (Z(I, {0, 1, }), τ(I,I)˙ ) is just a single spiral. The homeomorphism T (I, I) product of spirals indexed by the Cantor set C. • (A, A0 ) = (I, C) : We call (Z(I, C), τ(I,C) ) a line of spirals. Let x0 , x1 ∈ Z(I, C) with q(x0 ) ≤ q(x1 ) (Recall from above the map q : Z(I, C) → I obtained by ...
... ˙ is a (Z(I, {0, 1, }), τ(I,I)˙ ) is just a single spiral. The homeomorphism T (I, I) product of spirals indexed by the Cantor set C. • (A, A0 ) = (I, C) : We call (Z(I, C), τ(I,C) ) a line of spirals. Let x0 , x1 ∈ Z(I, C) with q(x0 ) ≤ q(x1 ) (Recall from above the map q : Z(I, C) → I obtained by ...
Topology 550A Homework 3, Week 3 (Corrections
... any straight line as a subspace of B is its usual topology. The relative topology on any circle in the plane as a subspace of B is the discrete topology. Proof . Let B denote the radial plane. Firstly, need to show that the relative topology induced on any straight line as a subspace of B is its usu ...
... any straight line as a subspace of B is its usual topology. The relative topology on any circle in the plane as a subspace of B is the discrete topology. Proof . Let B denote the radial plane. Firstly, need to show that the relative topology induced on any straight line as a subspace of B is its usu ...
Section 1.6-Classify Polygons
... Is a closed plane figure with the following properties: 1. It is formed by three or more line segments called sides 2. Each side intersects exactly two sides, one at each endpoint, so that no two sides with a common endpoint are collinear. ...
... Is a closed plane figure with the following properties: 1. It is formed by three or more line segments called sides 2. Each side intersects exactly two sides, one at each endpoint, so that no two sides with a common endpoint are collinear. ...