Tagged Questions
2
votes
1answer
35 views
A restricted continuous map is a homeomorphism
Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists $ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ ...
1
vote
1answer
54 views
Compactness and connectedness on $M_n(\mathbb R)$
Consider $M_n(\mathbb R)$, the set of all $n\times n$ matrices.
Which of the following are compact and which are connected?
a) The set of all invertible matrices
b) The set of all orthogonal ...
3
votes
2answers
67 views
Imposing the topology of open rays in $\Bbb R$
After having received Brian M. Scott's permission (see comments in the selected answer) I am integrating his suggestions with my own solutions to form a complete answer to the questions apperaing ...
3
votes
1answer
75 views
Minimal Connected Set containing a Closed Connected Set in a Compact Space
This question came from Dugundji's $\textit{Topology}$: Given a compact, connected space $X$, let $A \subset X$ be closed. Prove that there exists a closed, connected set $B \subset X$ such that $A ...
3
votes
1answer
81 views
Topological properties of symmetric positive definite matrices
Let $S$ be the set of all symmetric positive definite matrices of size $n\times n$. Which of the following statements are true?
(a) $S$ is closed in $\mathbb{M}_n(\mathbb{R})$.
(b) $S$ is ...
4
votes
2answers
109 views
Connectedness of $\beta \mathbb{R}$
I am not very familiar with Stone-Čech compactification, but I would like to understand why the remainder $\beta\mathbb{R}\backslash\mathbb{R}$ has exactly two connected components.
0
votes
0answers
35 views
How to prove that certain function with specific conditions is continuous [duplicate]
Possible Duplicate:
A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous
Let f be a function from $R^m$ to $R^n$. If image of every compact K is also ...
10
votes
2answers
792 views
Topology of matrices
1.Consider the set of all $n×n$ matrices with real entries as the space $\mathbb R^{n^2}$ .
Which of the following sets are compact?
(a) The set of all orthogonal matrices.
(b) The set of all ...
2
votes
0answers
102 views
Connectedness of the complement of a compact “small” subset of $\mathbb R^n$
Let $C$ be a compact subset of $\mathbb R^n$ and suppose that for every $\varepsilon >0$ there exists a finite family of open disks $B_i$ s.t. $C \subset \bigcup_{i} B_i$ and $\sum_i r_i \le ...
0
votes
0answers
144 views
Connected component
Read on topological spaces I found two doubts on the following result:
Let $T$ a compact and Hausdorff topological space and $C_t$ the connected component for $t\in T$.
a) Why if $T$ is compact ...
