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Project 2 is a writing assignment on examples satisfying topological definitions/properties

Project 2
Spring 2020
6 pages
Overview:
Project 2 is a writing assignment on examples satisfying topological definitions/properties. This
Project is to contain at least 1000 words. The project will contain topological definitions/properties
from class lectures and/or Hatcher’s notes (that is, the definitions/properties have be written
correctly and completely somewhere in the project). Topological definitions/properties are very
theoretical; so the main goal of the project is to write down examples of those topological definitions/properties.
In general, the writing needs to be grammatically correct, and the writing needs to adhere to the
conventions of written English, since the writing needs to be understood by the reader. In mathematical writing, the organization and development of ideas, clarity of expression, and coherence
between sentences and paragraphs are extremely important. The quality of the mathematical writing will be evaluated and used when establishing the graded for the project; in addition to the
topological definitions’ correctness and completeness, and topological examples’ correctness. The
~ Hire our professional writers now and experience the best assignment help online with our custom paper writing service. We ensure your essays and assignments are expertly researched, written and delivered on time. ~ Grading details will be given on the next page.
Three way to produce your final draft of your project:
• Type your Project using LATEX
• Type your Project using html with MathJax, or
• Legibly handwrite your Project.
NOTE: I will not except .doc style print outs! Legible handwriting is better than what .doc format
(and the like) can produce!
1
Math 314 Spring 2020: Project 2 page 2 of 6
Grade Evaluation
The grade earned on this project will be between .99 and .60.
The Rubric
In this section, we give more details (the fine print—the font is the same size as the rest of the
document, but whatever) as to how one earns grades in between .99 and .60.
Description (more details below)
15 points possible Lack of False Statements.
10 points possible Good Use of Grammar.
20 points possible From Examples.
4 points possible Good Style (very subjective).
First, sum the points earned in these four categories and 50 points; divide that sum by 100 points
to get a calculated number between .99 and .50 (no fraction of points will be given); second, the
earned grade is the maximum of that calculated number and .60 (to make sure the number is
between .99 and .60). See the examples below:
Say, a student earns the following points:
11 points is earned for a Lack of False Statements;
8 points is earned for Good Use of Grammar;
16 points is earned from Examples;
3 points is earned for Good Style (very subjective).
So we would do the following calculation
max
11 + 8 + 16 + 3 + 50
100
, .60
= .88,
which would be the grade earned for the student’s Project.
Math 314 Spring 2020: Project 2 page 3 of 6
Lack of False Statements
You will earn the following points for the lack of false written statements; that is, points are not
earned for the statements written, which are False (or True) but the writing communicates them
to be True (or False, respectively). Below is how these points are earned:
Earn Description
15 points for zero false statements found in the Project.
11 points for one false statement found in the Project.
7 points for two false statements found in the Project.
2 points for three to five false statements found in the Project.
0 points for more than five false statements found in the Project.
Good Use of Grammar
You will earn the following points for the quality of the grammar used in the project; that is, you
will not earn points, if the grammar used makes the written statements unclear to whether (or not)
the statement is True. Below is how these points are earned:
Earn Description
10 points for zero statements found where grammar make
their truth unclear.
8 points for one statement found where grammar make
their truth unclear.
6 points for two statements found where grammar make
their truth unclear.
4 points for three to five statements found where grammar
make their truth unclear.
0 points for more than five statements found where grammar make their truth unclear.
Math 314 Spring 2020: Project 2 page 4 of 6
The Examples
If the Project contains at least 1000 words then you will earn points for the Project containing
examples of the topological definitions/properties. Below is how these points are earned:
Earn Description
20 points for zero missed examples and/or topological definitions/properties.
16 points for one missed example and/or topological definition/property.
12 points for two or three missed examples and/or topological definitions/properties.
7 points for four or five missed examples and/or topological definitions/properties.
2 points for more than five missed examples and/or topological definitions/properties.
Good Style
You will earn points for the Project’s style; (from an English dictionary entry for “style” and
rewording with grammar) that is, the particular kind, sort, or type, as with reference to form,
appearance, or character of the Project is evaluated (this is very, very subjective). Below is to give
an idea how these points are earned:
Earn Description
4 points The typesetting (or handwriting) is very legible, is very organized, has high
clarity, is double spaced, makes it easier to understand, etc. (et cetera). Think
of what an awesome theoretical math textbook looks like.
3 points Not really as good as needed for 4 points, but better than what is needed for 2
points.
2 points The typesetting (or handwriting) is just legible, is organized, is double spaced,
etc. Think the typesetting (or handwriting) does not get “in the way” of the
reader’s comprehension.
1 point Not really as good as needed for 2 points, but better than what would receive
0 points.
0 points The typesetting (or handwriting) is not legible, is not organized, is not double
spaced, makes it so hard to understand it is not read, etc.
Math 314 Spring 2020: Project 2 page 5 of 6
Details about the Examples
First, below is a list of topological definitions/properties that one need to include in the project
(that is, the definitions/properties have be written correctly and completely somewhere in the
project). The project needs to include in the project the topological definitions/properties of the
following:
1. A topology O on a set X, a topological space X with topology O;
2. A open subset U of X with a topology O;
3. A closed subset C of X with a topology O;
4. The interior, closure, and boundary of a subset A of X with a topology O;
5. A basis B for a topology on a set X;
6. The two properties that a basis B satisfies (on pg 8 of Hatcher’s notes);
7. A metric d on a set X;
8. Open ball of radius r centered at the point x ∈ X;
9. A metric space;
10. A continuous function between two topological spaces;
11. The two lemmas about continuous function (on pg 12 of Hatcher’s notes);
12. A homeomorphism between two topological spaces.
Second, below is a list of the number of examples and of what topological definitions/properties
that one need to include in the project. The project needs to include (at minimum) the following
examples of the topological definitions/properties:
1. Six Xi with topology Oi (for i = 1, 2, 3, 4, 5, 6) examples which are topological spaces;
2. Six Ui (for i = 1, 2, 3, 4, 5, 6) examples that are open subsets of one example of a topological
space X with O;
3. Six Ci (for i = 1, 2, 3, 4, 5, 6) examples that are closed subsets of one example of a topological
space X with O;
4. Three subsets Ai (for i = 1, 2, 3) with the interior int(Ai), the closure Ai
, and the boundary
∂Ai from one example of a topological space X with a topology O;
5. Four Bi (for i = 1, 2, 3, 4) examples that are a basis for a topological space(s) (i.e. the
topologies can be different, just need four bases);
6. For two Bj (j ∈ {1, 2, 3, 4} of just above), show that Bj satisfies the two properties of a basis
(on pg 8 of Hatcher’s notes);
Math 314 Spring 2020: Project 2 page 6 of 6
7. Three functions di
: {(x, y) such that x, y ∈ Xi} −→ R that are metric on a set Xi (for
i = 1, 2, 3); here, the sets Xi
’s do not have to be different just the metrics need to be different;
8. For each of the three di on Xi (for i = 1, 2, 3 of just above), give three Brj
(xj ) examples
where each is an open ball of some radius rj centered at some point xj (for j = 1, 2, 3)—note:
for each i, there are three j’s; so nine total examples are needed;
9. For one of the di on Xi
, show the collection of all open balls
B = {Br(x) such that x ∈ Xi
, r ∈ R, r > 0}
satisfies the two properties of a basis (on pg 8 of Hatcher’s notes);
10. Three fi (for i = 1, 2, 3) that are continuous function between two topological spaces (i.e. the
topological spaces can be different for different i’s);
11. For one fi (for i ∈ {1, 2, 3} of just above), show that fi satisfies the two lemmas about
continuous functions (on pg 12 of Hatcher’s notes);
12. Give one homeomorphism between two topological spaces X with OX and Y with OY where
X and Y are different sets.

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