Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

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2
votes
1answer
32 views

Sequence in product metric space [on hold]

Let we have $(X_1,d_1)$ is a metric space and $(X_2,d_2)$ is another metric space . Now we will difend $X=X_1*X_2$ and we have $d$ is a distance function on $X$ So $(X,d)$ is a metric Space I ...
0
votes
1answer
14 views

$x \to e^{ix}$ , $[0,2\pi[$ onto $S_1$ is no a homeomorphisms

$f: x \to e^{ix}$ , $[0,2\pi[$ onto $S_1$, is no a homeomorphsims. Idea: We know $e^{ix} \leftrightarrow (\cos x,\sin x)$ onto $S_1 \subset \mathbb{R}^2$. Must be find a contraexample such that ...
2
votes
2answers
27 views

Rationale behind a proof regarding a continuous function and an open ball

can I have the rationale for the first line of this proof? i.e. How did you know to start answering the question in this manner? I am guessing it is because you want to exploit the definition of ...
1
vote
1answer
20 views

Show that there exists sets $A, B$ in $R$ such that $(A \cup B)^o \neq A^0 \cup B^o$

$\newcommand{\closure}{\operatorname{closure}}$ Show that there exists sets $A, B$ in $R$ such that 1) $(A \cup B)^\circ \neq A^\circ \cup B^\circ$ and $2)$ $\operatorname{closure}(A \cap B) \neq ...
0
votes
3answers
52 views

Simplest way to prove that $e^{ix}$ is an open mapping into $S^1$

Let $S^1$ be the unit circle in $\mathbb R^2$ and give it the subspace topology. What's the simplest way to prove that $f:R\to S^1$, $f(x)=e^{i x}$ is an open mapping, that is $f(U)$ is open when $U$ ...
0
votes
1answer
17 views

How do I show two different definitions for an exterior point are equivalent?

I have to show that the following two statements are equivalent. The first is the definition I was given by my professor for an exterior point. The second is the definition out of our textbook. ...
3
votes
2answers
58 views

If $Y$ is a dense second countable subspace of a regular space $X$, then $X$ is second countable.

Problem: Let $X$ be a regular space. If $Y$ is a subspace that is dense and second countable (with respect to the subspace topology), then $X$ is second countable. I have this problem and I'm having ...
-1
votes
1answer
34 views

Closure, interior, boundary and exterior of $A = (-\infty, 1) \cup [2, \infty)$ as a subset of $\mathbb R$ with the half open line topology

Let $ A = (-\infty, 1) \cup [2, \infty)$ be a subset of $\mathbb{R}$ in the half open line topology. Find each of the following: $\operatorname{Cl}(A)$ = I am totally unsure about this one? ...
1
vote
1answer
17 views

The distance of a point to a subset is greater than the distance of the subset to the boundary of the big set

Hi I'm trying to prove this that seems very obvious but I can't seem to prove it: if $V$ and $U$ are open sets (in $\mathbb{R^n}$) and $\overline{V} \subseteq U$ and $z \notin U$ then ...
1
vote
0answers
31 views

Quotient Space and Quotient Topology Definitions

I'm trying to show an equivalence between these two definitions: (1) The Quotient Space: Let $f: X \to Y$ be a map from a topological space $X$ to a set $Y$ and define $\pi: X \to \frac{X}{\sim}$ as ...
0
votes
2answers
28 views

Let $ X = \{a, b, c\}$ and $\mathfrak T = \{X, \emptyset, \{a,b\}, \{a\}\}$ Let $ A = \{a,c\}$

Let $ X = \{a, b, c\}$ and $\mathfrak T = \{X, \emptyset, \{a,b\}, \{a\}\}$ Let $ A = \{a,c\}$ I have to find each of the following sets and I think I am on the right track but I know I am not ...
4
votes
2answers
67 views

“$\sigma$-uniform continuity”

Let $X$ be an arbitrary metric space and $f:X\to\mathbb R$ a bounded continuous function. Is it possible to choose a countable sequence $(A_n)_{n\in\mathbb N}$ of (preferably open or closed) subsets ...
2
votes
0answers
66 views

Surgery to unlink $S^1$ and $S^2$ in $S^4$

Let us start with a $S^1$ and a $S^2$ are linked in $S^4$. Can I unlink the $S^1$ and $S^2$ by doing some surgery (with certain constraints described below, and let us say both $S^1$ and $S^2$ ...
1
vote
3answers
44 views

uniqeness of limit in topological spaces

let $(X,\tau)$ be a topological space such that for every $x\in X, \bigcap_{x\in U\in \tau}U= \{x\}$. Does that imply that every convergent sequence in $X$ has a unique limit?
0
votes
3answers
59 views

Closure of $A= (0,1) \cup (1,2)$ vs. Closure of $A = [0,1] \cup \{2\}$

Closure of $A= (0,1) \cup (1,2)$ vs. Closure of $A = [0,1] \cup \{2\}$ I am trying to figure out the difference of the closure of these two sets. Informally, my definition of closure is the ...
2
votes
1answer
26 views

Showing that a function is continuous

Let $A$ be a topological space, and let $B$ be a quotient space of it. I have defined a continuous function $F:A\times[0,1]\to B$ such that it factors into a function $G:B\times[0,1]\to B$, i.e. that ...
4
votes
2answers
72 views

Proof of $(0,1)$ is not compact with usual metric.

In the proof we say $\left\{\left(\frac1n,1\right):n\geq 1\right\}$ is an infinite cover with no finite subcover. But, $(0,1)$ set also belongs to cover mentioned above. We can say $\{(0,1)\}$ is a ...
3
votes
1answer
48 views

Under what conditions there exists a topology making a bijection into a homeomorphism?

Let $X$ be a set. Assume we have a bijection $f:X \rightarrow X$ which is not the identity. I want to find necessary or sufficient conditions on $f$ and $X$, that should imply existence of a ...
4
votes
3answers
42 views

Prove A is an open set if and only if $A \cap Bd(A) = \emptyset $

Prove A is an open set if and only if $A \cap Bd(A) = \emptyset $ Here is my start: Suppose A is an open set. We know $X-A$ is closed. Need to show $A \cap Bd(A) = \emptyset$ Let $ x \in A$. ...
1
vote
1answer
20 views

Prob. 6, Sec. 21 in Munkres' TOPOLOGY, 2nd ed: How to show directly that this sequence of functions does not converge uniformly?

For each $n = 1, 2, 3, \ldots$, let $f_n \colon [0,1] \to \mathbb{R}$ be defined by $$f_n(x) \colon= x^n \ \ \ \mbox{ for all } \ x \in [0,1].$$ Then $$ \lim_{n \to \infty} f_n(x) = \begin{cases} ...
4
votes
0answers
27 views

Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R}$ is closed and connected

Are there a closed connected subspace $C$ of $\mathbb{R}^2$ and a continuous, bijective function $f:C\to\mathbb{R}^2$ that is not open? If we remove the condition for $C$ to be connected, we have ...
2
votes
0answers
47 views

How to define the boundary operator using the exterior derivative?

I am looking for a way to define the boundary operator $\partial : M^n \to N^{n-1}$ from an $n$-dimensional manifold $M$ to its boundary $N$ using the the expression \begin{equation*} \int_M d \alpha ...
0
votes
1answer
31 views

Prove that, in the usual topology, both $a$ and $b$ are in the boundary of each $(a,b)$ and that no other point is in the boundary.

Suppose that $a$ and $b$ are real numbers such that $ a \lt b$. Prove that, in the usual topology, both $a$ and $b$ are in the boundary of each $(a,b), [a,b], [a,b),$ and $(a, b]$, and that no other ...
1
vote
0answers
17 views

A calculation for an open ball in $\mathbb{R}^N$ and function space.

Let $B_r(x)$ denoting the ball of center $x$ and radius $r>0$. We denote by $\lambda_{1,\,B_\rho(y)}$ the first eigenvalue of $-\Delta$ in $W^{1,\,2}_0\left(B_\rho(y)\right)$ and by ...
1
vote
1answer
37 views

Injection of the mapping cone of $z^2$

We define the mapping cone of $f:S^1\to S^1=:Y$, $f (z)=z^2$ as the quotient space of $S^1\times [0,1]\sqcup Y$ where $(z,0)$ and $(z',0)$ are identified and where $(z,1)$ and $f(z)$ are identified ...
1
vote
1answer
25 views

Proving isometry and continuity from a positive definite symmetric real matrix

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\epsilon$ be the Euclidean metric on $\Bbb R^n$ ...
3
votes
1answer
32 views

Show that a sequence (($x_n, y_n$)) in $X \times Y$ is $e$-Cauchy if the component sequences ($x_n$) and ($y_n$) are $d_X$-Cauchy and $d_Y$ -Cauchy.

How to solve this? Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces and let $e$ be a product metric on $X\times Y$. Show that a sequence (($x_n, y_n$)) in $X \times Y$ is $e$-Cauchy if the component ...
2
votes
1answer
23 views

Surjectivity of expanding map

Suppose that $(X, d)$ is a compact metric space and that $f: X \rightarrow X$ is a continuous function satisfying $d(x,y) \leq d(f(x), f(y))$ for all $x, y \in X$. Show that $f(X) = X$. Here is a ...
1
vote
1answer
29 views

Proving the continuity of functions from one metric to another

I'm studying mathematics at university, and am having trouble with some of the continuity questions. The following is a question from a previous assignment that I was unable to complete. The original ...
1
vote
1answer
32 views

Quotient of union of two spaces

Let $X$ be a topological space, $f : S^{n-1} \to X$ and $Y := X \cup_f D^n = \big(X \coprod D^n\big) / \sim$ , where $t \sim f(t)$ for $t \in S^{n-1}$. Problem. Prove that $Y/X \cong S^n$. My idea. ...
0
votes
0answers
21 views

Prove every bounded sequence in the real numbers has a convergent subsequence using cauchy

Prove that every bounded sequence in $\mathbb{R}$ has a convergent subsequence. I am going to use the Cauchy sequence property to do this. Proof: Let $A_n$ be a bounded sequence in $\mathbb{R}$. ...
0
votes
0answers
22 views

Proving that if the real numbers are complete than every cauchy sequence converges

Prove that if $\mathbb{R}$ is complete then every Cauchy Sequence converges. Definition of complete: A set $A$ of real numbers is said to be complete if every Cauchy sequence $<a_n\in ...
2
votes
1answer
19 views

Prove that identity map f from (X,T) (topological space) to (X,T') (cofinite topological space) is continuous if (X,T) is a T1 space

So following is the proof that I have come up with but I'm not sure if it's correct or not. I'd be really grateful if you could validate/invalidate it. To prove: Given $(X,T)$ is $T_1$, prove that ...
1
vote
1answer
35 views

Showing that the center of a Möbius strip is homeomorphic to a circle?

Consider the Möbius strip as the quotient space obtained from $[0,1]\times[0,1]$ when identifying $(0,t)$ with $(1,1-t)$ for $0\leq t\leq1$. Now consider its center, that is, the image of the set ...
1
vote
0answers
32 views

quick question about a definition of homeomorphism set classes

Take $S$ a surface of general type. I want to define $Q$ the set of homeomorphism classes determined by the surface $S$. How can i define $Q$? I think that $Q$ is determined by all surfaces $S^{'}$ ...
3
votes
1answer
49 views

Difference between Euler characteristics of a Riemann surfaces

Let $X$ be a compact connected Riemann surface of genus $g$. Let $U$ be the complement of $r$ points in $X$. The Euler characteristic of $X$ = $2-2g$. That I understand. But I'm confused about the ...
1
vote
1answer
26 views

Between metacompactness and infinite Lebesgue covering dimension

A space $(X,\tau)$ is said to be metacompact if every open covering $\cal U$ has a refinement $\cal V$ such that for every $x\in X$ the set ${\cal V}_x := \{V\in \mathcal{V}: x\in V\}$ is finite. A ...
7
votes
2answers
353 views

Existence of a continuous function which does not achieve a maximum.

Suppose $X$ is a non-compact metric space. Show that there exists a continuous function $f: X \rightarrow \mathbb{R}$ such that $f$ does not achieve a maximum. I proved this assertion as follows: ...
4
votes
1answer
44 views

Zero-dimensional but not Hausdorff

Let's call a space zero-dimensional if it has a basis of clopen sets, and is $T_0$. Is there a zero-dimensional space that is not Hausdorff?
1
vote
2answers
46 views

$C$ is an uncountable set. Show that $B(x,\epsilon)\cap C$ is uncountable.

Suppose $X$ is a separable metric space, and $C$ is an uncountable subset of $X$. Prove that there is a point $x \in C$ such that for each $\epsilon>0$, $B(x;\epsilon)\cap C$ is uncountable. I ...
1
vote
1answer
31 views

Continuous function in order topology

I have to prove that for every $ \alpha$ in $[0, \Omega)$, there exists a continuous function $f$ from $[0, \Omega)$ to $\mathbb R$ such that the pre-image of $0$ is $\{α\}$. I really have no idea ...
7
votes
3answers
113 views

Some topologies are more equal than others

On MathOverflow 5 years ago, I answered a question about Awfully sophisticated proofs for simple facts. I answered Fürstenberg's topological proof of the infinitude of primes. While the answer ...
0
votes
1answer
14 views

Lower semicontinuous non-negative function on a locally compact Hausdroff space with a countable base

An extended real number is an element of $\mathbb R \cup \{-\infty, +\infty\}$. Let $X$ be a locally compact Hausdorff space with a countable base. An extended real valued function $f$ on $X$ is ...
0
votes
1answer
22 views

What can I say about $f:G \to Y$?

If $f:X \to Y$ be inyective where $X$ is nonempty perfect Polish space, $Y$ second countable and exist $G\subseteq X$ such that $G=\bigcap_{n \in \omega}D_n$ where $D_n$ is dense open sets and ...
1
vote
0answers
41 views

Pushout in $\mathsf{Set}$ where one of the maps is injective

From I.M. James' book General Topology and Homotopy Theory: Suppose we have a cotriad $$X \xleftarrow{\xi}W \xrightarrow{\eta} Y.$$ ... we might expect the pushout of the cotriad to be a ...
3
votes
1answer
87 views

Show that $\bar A = A \cup [(0,0), (0,1)]$

In $(\mathbb R^2, ||\cdot||_{\infty})$, let: $A_0 = ]0,1] \times \{0\}$ $A_n = [(\frac{1}{n}, 0), (\frac{1}{n}, 1)]$ for each $n \ge 1$. $A = \cup_{n=0}^{\infty} A_n$ It is required to prove that: ...
3
votes
1answer
56 views

projective space and torus

we defined the projective space as $\mathbb{S}^2$ with opposie side identification and the torus as $\mathbb{R}^2 / \mathbb{Z}^2.$ And now I am concerned with their manifold structure- In fact, I ...
0
votes
1answer
43 views

Hahn-Banach Thm for Normed Space [closed]

Let $X$ be a normed Space. For $x \in X$ define $J(x)=\{f \in X^ * : f(x)=\|x\| , \|f\|=\|x\|\}$. Prove that $J(x)$ is not the empty set.
1
vote
0answers
27 views

Lie bracket question

I am wondering if this is correct. Suppose $X$ and $Y$ are two smooth vector fields which vanish at $p$: $X(p) = Y(p) = 0$. Also assume that $[X, Y](p) = 0$. Is it true that the derivative of the ...
3
votes
0answers
63 views

Am I a toroid or not?

I heard that topologically (whatever that is, me not good at math), a donut and a mug are the same thing. Both have a hole going all the way through them. I found this animation transforming one into ...