This is a closed-book, closed-notes exam. You have one hour to complete
the test. Write your answers in the space provided. You may use blank paper
for scratchwork or as additional space for solutions.
After you have completed your work, sign the statement below.

I accept full responsibility under the Binghamton University Academic
Honesty Code for my conduct on this examination.
Signed:

## Math 461, Fall 2021

Each problem is worth 5 points.

Problem 1. Definitions: Provide definitions of the following terms.

1. What properties must a set $X$ and a function $d: X \times X \rightarrow \mathbb{R}$ satisfy to be a metric space?
2. Given a topological space $X$ with an equivalence relation $\sim$, what is the quotient topology on $[X] ?$4

Problem 2.

True/False: Determine whether each statement below is True or False. Justify your answer as clearly and completely as you can.

1. Every metric space is Hausdorff.

Problem 3.

1. If $x$ is a limit point of $A \subset X$ then $x \notin A$

Problem 4.

1. If $f: X \rightarrow Y$ is continuous and $C \subset X$ is closed, then $f(C)$ is closed.

real analysis代写analysis 2, analysis 3请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。

# Binghamton University Graduate Combinatorics, Algebra, and Topology Conference (BUGCAT)

The 14th Annual Binghamton University Graduate Combinatorics, Algebra, and Topology Conference (BUGCAT) is to be held online through Zoom, November 6th, 7th, 13th, and 14th 2021.

This year’s featured keynotes are Profs. Tara Holm from Cornell University, Dandrielle Lewis from High Point University, and Inna Zakharevich from Cornell University.

Visit the conference home page and the conference program page. There is also a Facebook Page, or you may e-mail [email protected] for more information.

Categories: 拓扑学数学代写