MY-ASSIGNMENTEXPERT™可以为您提供gvsu.edu MTH441 Topology拓扑学的代写代考和辅导服务！
这是大峡谷州立大学拓扑学课程的代写成功案例。
MTH441课程简介
An introduction to the fundamental concepts of topology. The topology of the real number system and its generalizations to metric spaces and topological spaces. Topics include subspaces, neighborhood spaces, open and closed sets, interior and boundary of sets, continuity and homeomorphisms, connected and locally connected spaces, compact sets and spaces. Prerequisites: MTH 203, MTH 210, and MTH 204.
Credits: 3
Prerequisites
Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid. Similarly, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle (i.e., a one-dimensional closed curve with no intersections that can be embedded in two-dimensional space), the set of all possible positions of the hour and minute hands taken together is topologically equivalent to the surface of a torus (i.e., a two-dimensional a surface that can be embedded in three-dimensional space), and the set of all possible positions of the hour, minute, and second hands taken together are topologically equivalent to a three-dimensional object.
MTH441 Topology HELP（EXAM HELP， ONLINE TUTOR）
Let $\Gamma \subset \mathbf{R}^2$ be the graph of a function $f: \mathbf{R} \rightarrow \mathbf{R}$. If $\mathbf{R}$ and $\mathbf{R}^2$ are given the standard topologies (and as we now know, the standard topology on $\mathbf{R}^2$ is the same as the product topology) and if $f$ is continuous with respect to the standard topologies, show that $\Gamma$ is closed. Hint: Define $F: \mathbf{R}^2 \rightarrow \mathbf{R}$ by $F((a, b))=f(a)-b$. Show that $F$ is continuous and that $\Gamma=F^{-1}({0})$. You may assume that $+: \mathbf{R}^2 \rightarrow \mathbf{R}$, defined as $+((a, b)) \mapsto a+b$, is continuous, and that $-: \mathbf{R} \rightarrow \mathbf{R}$, defined as $-(a)=-a$, is continuous.
Let $\pi_2: \mathbf{R}^2 \rightarrow \mathbf{R}$ be defined by $\pi_2((x, y))=y$. If $\mathbf{R}$ and $\mathbf{R}^2$ are given the standard topologies, show that $\pi_2$ is not a closed map (i.e., give an example of a closed subset $C \subseteq \mathbf{R}^2$ such that $\pi_2(C)$ is not closed).
Give an example of a relation $R$ on a set $S$ which is reflexive and symmetric but not transitive.
Give an example of a relation $R$ on a set $S$ which is reflexive and transitive but not symmetric.
Give an example of a relation $R$ on a set $S$ which is transitive and symmetric but not reflexive.
As we defined in class, a short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$, with maps $i: A \rightarrow B$ and $j: B \rightarrow C$ splits if there is an isomorphism $\alpha: B \rightarrow A \oplus C$ with $\alpha \circ i$ being inclusion in the first factor and $j \circ \alpha^{-1}$ being projection onto the second. Prove that the sequence splits if (and only if) there exists a homomorphism $\ell: B \rightarrow A$ such that $\ell \circ i$ is the identity on $A$.
Prove the 5-lemma. Yes, we did this in class, but writing out the details of this sort of diagram-chasing will give you a better feel for the technique. Of course you can just copy the notes you (possibly) took in class, and you’re free to do so, but please try to prove it on your own first.
MY-ASSIGNMENTEXPERT™可以为您提供GVSU.EDU MTH441 TOPOLOGY拓扑学的代写代考和辅导服务！
