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
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1answer
84 views

Give example of $f$ that is open but neither closed not continuous (in 2D).

I'm trying to teach my self topology. The book I'm using has the following problem: Give an example of two subsets $X,Y \subseteq \mathbb R ^2$, both considered as topological spaces with their ...
2
votes
1answer
33 views

Example of Kolmogorov space which is neither Fréchet nor sober

Is every Kolmogorov ($T_0$) space either Fréchet ($T_1$) or sober or both, or is it possible to be Kolmogorov but neither of those two? If the latter, can a counterexample be defined? A sober space ...
1
vote
0answers
32 views

Proving sigma-algebras equality

I'm not sure if my proof of the following statement is correct. Let $\tau : \Omega \to \mathbb{R}_+$ be a non-negative random variable, defined on a probability space $(\Omega, \mathcal{G}, \...
-2
votes
1answer
34 views

Meaning of measurable function [closed]

I don't understand the meaning of Measurable function , my lecture told us that $f(x)$ measurable on measurable set $E$ if $E(f>A)=\{x\in X:f(x)>A\}$ $1.\quad$Can you please give me examples ...
1
vote
2answers
44 views

Equivalent definitions of continuity at a point

I'm going with a definition of a map over defined on topological spaces $f:X\rightarrow Y$ is continuous at a point $x\in X$ is as follows: $f$ is continuous at each element $x\in X$ if and only ...
0
votes
0answers
20 views

Does Stone's theorem works with a pseudometric?

Basically I don't know if a pseudometric space is paracompact. I thought in the Stone's theorem but Stone requires a metrizable space, but I check the demostration and which does not use strongly ...
1
vote
0answers
61 views

What kind of morphisms should I expect to be proper?

I'm trying to learn more about proper pushforward, but I'm stuck at coming up with interesting examples of proper morphisms of schemes. The only examples I can think of are inclusions of projective ...
4
votes
3answers
241 views

Question about continuity in the box topology

I have two question regarding the following example in Munkres (1)Why "if $f^{-1}(B)$ were open it would contain some interval $(-\delta,\delta)$ about 0. (2)My second question is somewhat broad, but ...
1
vote
1answer
30 views

Show a set $C$ is closed iff point to set distance is not zero for all points outside of the set

I think this is an interesting problem Suppose that we have $X$ with the metric topology $\tau$ such that $C$ is a closed set in $\tau$ Define the point to set distance as $$dist(x, C)...
3
votes
0answers
139 views

Is $X\simeq [0,1]$?

Suppose that $X$ is a metric continuum irreducible between two points $p$ and $q$. Suppose further that whenever $U$ is a connected open set missing $p$ and $q$, we have $X\setminus U$ has two ...
2
votes
1answer
35 views

Show that any metrizable space $X$ is Hausdorff

I wish to show that any metrizable space $(X,\mathcal{T})$ is Hausdorff Proof attempt: Let $d$ be the metric that generates the topology on $X$. Pick two points $x,y \in X$, we wish to produce two ...
2
votes
0answers
34 views

Show that if $(X, \mathcal{T}), (Y, \mathcal{J})$ are both metrizable, then $X \times Y$ with product topology is metrizable

Let $(X, \mathcal{T}), (Y, \mathcal{J})$ be both metrizable, then Claim: $X \times Y$ with product topology is metrizable I have made an attempt at this question but the notation is ...
1
vote
0answers
31 views

Contractions mappings bijective maps boundarys on boundarys?

I remenber here the concept of a contraction mapping. Definition: Let (X, d) be a metric space. Then a map T : X → X is called a contraction mapping on X if there exists q ∈ [0, 1) such that $$ d(T(...
0
votes
2answers
31 views

Zeros and poles of some meromorphic 1-forms on the riemann sphere

Let $X=\mathbb C_{\infty}$ be the Riemann sphere with the local coordinates $\{z\ ,1/z\}$. I want to show the following two statements: i) There does not exist any non-vanishing holomorphic 1-form on ...
1
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0answers
24 views

A non-constant holomorphic map $F$ between riemann-surfaces is an isomorphism

I want to show the following: Let $F:X\rightarrow Y$ be a non-constant and holomorphic map between compact riemann surfaces with $genus(X)=genus(Y)\geq 2$. In the above it holds that $F$ is an ...
1
vote
1answer
43 views

General topology question-simply connected, connected boundary

Suppose $\Omega\subset\mathbb{R}^3$ be a bounded connected open set. Then $\Omega_C\subset\Omega$ is an open set such that $\overline{\Omega_C}\subset\Omega$ and such that $\Omega_I:=\Omega\setminus\...
0
votes
1answer
29 views

Prove $(\text L)\int_0^1[x-\text K(x)]\sin x\text d x= (\text L)\int_0^1x\sin x \text dx$

Let $\text K(x)$ be a Cantor function on $[0,1]$ prove $$(\text L)\int_0^1[x-\text K(x)]\sin x\text d x= (\text L)\int_0^1x\sin x \text dx$$ here $(\text L)$ denotes Lebesgue-integral. Attempt: ...
0
votes
1answer
19 views

Helly theorem application

Let $X$ be a normed space, $dim(X)=d$, let $r>0, A$ and $\subset X$ . Show that if every $d+1$ points of $A$ are contained in a closed ball of radius $r$, then $A$ is contained in a closed ball of ...
1
vote
2answers
40 views

Box topology definition

Munkres defines the box topology as shown below. I am trying to understand how come an element of basis as defined is even subset of $\prod_{\alpha \in J}X_{\alpha}$. If we get an element of the basis,...
1
vote
1answer
32 views

is the function $\rho$ a pseudometric?

Let $\Im=\left\{\Im_{n}\right\}_{n\in \mathbb{N}}$ be a sequence of open covers of a topological space $(X,\tau)$. We define a function $\delta:X\times X \rightarrow \mathbb{R}$ as follows If $(x,y)\...
1
vote
1answer
26 views

Rules for constructing continous functions

In the proof below I don't understand the statement. Because Z contains the entire image set f(X), $f^{-1}(U) =g^{-1}$, by elementary set theory.
6
votes
2answers
201 views

If $K$ is compact and $C$ is closed in $\mathbb{R}^k$, prove that $K + C$ is closed using a “direct” proof

Rudin Exercise 4.25(a) reads: If $K$ is compact and $C$ is closed in $\mathbb{R}^k$, prove that $K + C$ is closed. The hints in the problem suggest a proof by proving that the complement of $K + C$ ...
2
votes
2answers
46 views

Is every generalized-$F_{\sigma}$ set an $F_{\sigma}$ set?

A subset $S$ of a topological space $X$ is called a generalized-$F_{\sigma}$ set in $X$ if for all open $G \subset X$ with $S\subset G$, there exists an $F_{\sigma}$-set $F$ such that $S\subset F\...
2
votes
0answers
39 views

Finding closure of image of operator

I'm working on an old exam problem: Define for $u \in C^2([-1,1])$ the operator $L$ by $[Lu](x) = - \frac{d}{dx} \left( (1-x^2) u'(x) \right)$. Set $\Omega = \{ Lu \mid u \in C^2([-1,1]) \}$. Find the ...
1
vote
1answer
35 views

Topology given by atlas is paracompact

I'm currently reading Jeffrey M. Lee Manifolds and Differential Geometry book. I don't understand a part in the proof of Proposition 1.32. (iii). Proposition 1.32. says: Let $M$ be a set with a $...
-1
votes
1answer
42 views

Hausdorff compact with closed function [closed]

Let $X$ be a compact Hausdorff space and let $f$ be continuous, closed, surjective function from $X$ to $Y$. Prove that $Y$ is Hausdorff. It is obvious that $Y$ is Hasudorff space. However, I have ...
0
votes
1answer
95 views

$(0,1) $ with the usual topology admits a metric which is complete?

Is the following statement is true? $(0,1)$ with the usual topology admits a metric which is complete? My answer is "False." But, the answer given is "True". I am unable to figure out. Please help ...
1
vote
1answer
20 views

Which subsets of products are graphs of continuous maps?

Let $f:X\to Y$ be a map between topological spaces, and let $\Gamma_f\subseteq X\times Y$ be its graph $\{(x,f(x))\mid x\in X\}$. Is there a (general) predicate $P(S',S)$ on pairs with $S'$ arbitrary ...
0
votes
1answer
33 views

Proof idea: Let $(X,d)$ be a metric space, and $\rho$ be bounded metric, show that they will generate the same topology

Let $(X,d)$ be a metric space, $d$ generates the metric topology $\mathcal{T}$ via metric ball $B_\epsilon(x)$. Show that bounded metrics: $\rho_1(x,y) = \dfrac{d(x,y)}{1+d(x,y)}$ with ...
0
votes
2answers
42 views

Show that $d_1=\min(d(x,y),2)$ is a metric space [duplicate]

Show that $d_1=\min(d(x,y),2)$ is a metric space if it is given that $d(x,y)$ is a metric space. I am stuck at the triangle inequality part, to show that $d_1(x,z)\leqslant d_1(x,y)+d_1(y,z)$ i.e ...
0
votes
5answers
61 views

$x\in \bar{A}$ iff every neighbourhood of $x$ intersects $A$.

I read the proof of this theorem from Munkres, however I don't really understand intuitively why this is true. If someone could provide me of intuition of this theorem that would be nice. $\bar{A}$ is ...
0
votes
1answer
25 views

Compact Hausdorff space with closed set

Suppose $X$ is a compact Hausdorff space and $f$ from $X$ to $X$ a one-to-one continuous function. Show that there exist a nonempty closed subset $A$ of $X$ such that $f[A] = A$. It is obvious that $...
0
votes
2answers
38 views

Let $X$ be a non-empty compact Hausdorff space. Which of the following statements are true?

Let $X$ be a non-empty compact Hausdorff space. Which of the following statements are true? $a.$ If $X$ has at least $n$ distinct points, then the dimension of $C(X)$, the space of continuous real ...
0
votes
1answer
38 views

Is there a topology which is coarser than the product topology?

Is there a topology which is coarser than the product topology on an infinite Cartesian product of topological spaces?
0
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1answer
26 views

Inequality in chapter “pontryagin construction” from Milnor's topology from the differentiable viewpoint

at the moment I am reading Milnor's book "topology from the differentiable viewpoint". I am stuck on page 49 (chapter "the pontryagin construction", proof of lemma 4). There he claims, that for $0<|...
0
votes
3answers
63 views

Prove a set to be an open set

Let $f:R\rightarrow R$ be a continuous function ,and let $U=\left\{(x,y):y>f(x)\right\}$ Prove that U is an open set in $R^{2}$. The result is intuitively obvious.My thought is following: for any ...
1
vote
0answers
17 views

Interior algebra vs. Regular open algebra

I know that an interior algebra is a Boolean algebra on which is defined an operation satisfying Kuratowski's axioms for the interior of a set. I recently heard about the regular open algebra, which ...
2
votes
0answers
29 views

Homotopy of closed curves is also a closed curve?

I'm trying to prove the following statement: Let $\gamma_1$ and $\gamma_2$ be two closed curves from $[a, b]$ to $\mathbb{C}$, and let $h: [a,b]\times[0,1] \to \mathbb{C}$ be a homotopy between ...
2
votes
1answer
29 views

a counter example of extension of a continuous function

Let $A\subset X$. Suppose $f:A\rightarrow Y$ is continuous, where $Y$ is Hausdorff. Show that if $f$ can be extended to a continuous function $g:\overline {A}\rightarrow Y$, then $g$ is uniquely ...
1
vote
0answers
75 views

A question on connectedness

I was going through a basic problem book in algebraic geometry. There in the very first chapter I have encountered a problem which asks to prove that hyperbolas are connected in $\mathbb{C}\times \...
2
votes
2answers
27 views

To find two points in compact metric space satisfying specific property

Let $(X,d)$ be a compact metric space.Suppose that for all positive real numbers $t<1$ ,there are points $x_{t}$,$y_{t}$ such that $d(x_{t},y_{t})=t$.Prove that there are points $x$, $y$ in $X$ ...
0
votes
1answer
28 views

Product of perfectly-$T_3$ spaces

This is (sort of) a continuation of this question Let $X$ be a topological space. Then, we say that $X$ is perfectly-$T_3$ iff it is $T_1$ and for every $x\in X$ and closed subset $C\subseteq X$ not ...
0
votes
1answer
24 views

Which of the following subsets of $M_n(\mathbb C)$ are compact?

Let $A ∈ M_n(\mathbb C)$ and let $\rho (A) = \max \{| \lambda| : \lambda$ is an eigenvalue of $A\}$ denote its spectral radius. Which of the following subsets of $M_n(\mathbb C)$ are compact? $a. S ...
1
vote
3answers
51 views

A canonical example for spaces that aren't $1^{\text{st}}$ countable

To get the feeling for metric spaces (or $1^{\text{st}}$ countable spaces, in general), I always find that visualizing them as $\Bbb R^3$ gives a good intuition of what is going on (locally, of course)...
3
votes
1answer
44 views

$K = \{A ∈ M_n(R) \mid A = A^T , \operatorname{tr}(A) = 1, x^T Ax ≥ 0 \ \ \forall x ∈ R^n\}$. Then $K$ is compact.

Let $K ⊂ M_n(\Bbb R)$ be defined by $$K = \{A ∈ M_n(\Bbb R) \mid A = A^T , \operatorname{tr}(A) = 1, x^T Ax ≥ 0 \ \ \forall x ∈ \Bbb R^n\}$$ Then $K$ is compact. Considering the continuous map $A \...
6
votes
1answer
88 views

what is the nature of a ball that goes over a “corner” of the real projective plane?

I'm make a little computer program to help me understand different 2d topological spaces, (such as torus and mobius band). I'm having issues with drawing balls that go over a corner of the real ...
1
vote
0answers
45 views

Three Jordan curves made from paths

For continuous paths $\phi,\psi:[0,1]\to \mathbb{R}^2$ such that $\phi(1)=\psi(0)$, let $\phi*\psi$ denote the composition path and $\phi^{-1}$ the inverse path. Consider points $p,q\in\mathbb{R}^2$ ...
1
vote
0answers
35 views

Surjectivity on the image of a annulus

I'm trying to prove the Fundamental Theorem of Algebra as it is done in Birkhoff and MacLane. Unfortunately, I don't have access to the book, only to a sketch. Therefore, I'm filling the gaps myself. ...
1
vote
0answers
48 views

metric space with no perfect set

Let $X$ be a complete separable metric space containing no perfect set of size greater than $1$. In other words every subset of $X$ has an isolated point. It is well known that $X$ must be countable....
1
vote
1answer
18 views

Union of Interiors is Subset of Interior of Union

I'm teaching my self topology using a book I found. This is the forth part of a 4 part question. links to other parts: one, two, three . I'm trying to prove the following problem from a book I found: ...