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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Can there be weak* open cover of the dual banach space with the arbitrary small (in diameter) sets?

That is, I want to cover $X^*$ (X is Banach space) with a family $\{U_{\alpha}\}$, where $diam(U_{\alpha})<\epsilon$ and each $U_{\alpha}$ is weak* open. I expect, that not every open ball is ...
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52 views

Number of topologies on a set

Let $X$ be a nonempty set with $n$ elements. I want to find an upper bound for the number of possible topologies for $X$. I proceed as follows: The power set $\mathcal P(X)$ contains $2^n$ elements. ...
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63 views

A counter example

If a set is compact in $Z(\mathbb{A})\setminus GL(2,\mathbb{A})$,then can it be compact in $GL(2,\mathbb{A})$ ? ($\mathbb{A}$, is the adele ring of $F$ on which $GL(2)$ is and $Z$ is the center of ...
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38 views

Importance of metrization theorem?

I wonder if there is a case metrization theorems(such as Nagata-Smirnov, Bing, Urysohn) pave a way to do a theory. What would be a nice application of metrization theorems?
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48 views

$E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering

Let $q:E\to X$ be a covering map. Then $E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering. My question is regarding the $"\implies"$ direction: If $E$ is compact, then ...
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50 views

Showing that an evaluation map is continuous

This is a problem from Munkres' Topology 43.8 If $X$ and $Y$ are spaces, define e : $X \times \mathscr {C}(X,Y) \to Y$ by the equation e($x,f$) $= f(x)$; the map e is called the evaluation map. ...
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34 views

No norm consistent with given topology

Given the (Frechet) topology on the Schwartz class $S(\mathbb{R}^d)$ induced by the seminorms $\rho_{\alpha \beta}f = \operatorname{sup}_{x \in \mathbb{R}^d}|x^{\alpha}\partial^{\beta}f|$, how can I ...
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56 views

If $X$ and $Y$ are homotopic and $X$ is contractible, so is $Y$

I want to show that if $X$ and $Y$ are homotopic and $X$ is contractible, so is $Y$. It feels like I'm missing something really obvious. $X$ is homotopic to $Y$, so there exists $f: X \to Y$ and $g: Y ...
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59 views

Number of connected components of $f^{-1} (U)$

Let $f:\mathbb{R}^n \to \mathbb{R}$ be an analytical function (semialgebraic,polynomial if needed), $U$ be an open connected subset of $\mathbb{R}$. What can we say about the nuber of connected ...
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59 views

Can you deformation retract a sphere to a point?

So, I'm working on a topology problem (Calculating the fundemental group of two spheres adjoined by a single point). As a subpart of the problem, we're trying to figure out if a sphere by itself can ...
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34 views

Proof of Version of Krasner's Lemma

I need some help in constructing the proof to this version of Krasner's Lemma from Serre's Local fields text book: Let E/K be a finite Galois extension of a complete field K. Prolong the valuation ...
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37 views

Degrees of spaces of polynomials

Let $I$ be an ideal in $K[x_1,\dots,x_n]$ where $K$ is a char $0$ field. Let $Z(I)$ be a set of discrete points whose cardinality is exponential in $n$ and spanning $n$ dimensions. Let $P$ be the ...
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69 views

Isomorphism for the group of units of the ring of integers of a local field

Let $K$ be a local field with a discrete and non-archimedean absolute value, $\mathcal{O}_K$ be its ring of integers, $\mathfrak{m}_K$ be the unique maximal ideal of $\mathcal{O}_K$ and ...
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53 views

What's the general technique to show a sequence converges?

After "guessing" what the limit of a particular sequence is, what's the general process to prove that this sequence indeed converges to it? (using the definition) (The definition says that a sequence ...
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31 views

Proof of Theroem 2-7 in Rudin's Real and Complex Analysis

I am working on the proof of Theorem 2-7 in Rudin's Real and Complex Analysis and I need some help clarifying why p does not belong to the closure of Wp. Here is a link to the entire theorem. ...
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52 views

Prove that this space is contractible

Consider the space $X/\sim=((-1,1)\times \{0,1\})/\sim$ where $(x,0) \sim (x,1)$ for $x\leq0.$ Prove that $X/\sim$ is contractible. I suppose we can contract $X/\sim$ to the point $[(0,0)] = ...
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13 views

use picture to show the conjugate of a loop in the fundamental group of a space

Give an example of a space in which a loop has a conjugate that is not equivalent to the original loop. this is another question but we did not need work on this
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60 views

What does $R^I$ stand for?

In section 30 of Munkres, one exercise states that "Give $R^I$ the uniform metric, where $I=[0,1]$". I guess it's not about powers or something, it's some conventional notation because I've never ...
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80 views

Lindelöf Property and Compact space

Let $X$ be a compact space and $L$ is the smallest family of subspaces of$\,X\,$that contains all closed sets and is closed with respect to countable union and intersection. The question is :- Is ...
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40 views

Levels sets of a continuous function

Suppose $f:[0,1]\rightarrow [0,1]$ is continuous. Let $A$ be the set of all maximal, connected subsets of the level set $f^{-1}(0)$. Can $A$ be uncountable?
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57 views

How is this topological space different from the euclidean one?

I'm preparing for my topology exam and came across this example which I can't figure out. Let $\mathcal{T}$ be a such family of all sets $U\subset \mathbb{R}^2$ that $U\cap L$ is an open set in L, ...
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27 views

Is the translation of open and closed sets to some language non-antonym preserving?

Maybe more than one person though, before you were given the definition of closed set, that they were the sets that are not open, i.e. that the property of open and closed being antonyms were ...
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108 views

Is this a general structure for constructs?

Here a construct is a category where the objects are sets and the morphisms are structure preserving functions. Common examples are groups, graphs and topological spaces. As far as I can see there is ...
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36 views

When is a metrizable topological vector space locally bounded?

Consider a topological vector space $E$ with topology $\sigma$. Suppose that $E$ is metrizable, in other words, that there exists a metric $d$ on $E$ that induces the topology $\sigma$. One can then ...
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38 views

An excerpt from a seminar

It is a statement that a professor made in a seminar which I attended yesterday.He says that the following hold: $1$.If $D$ denotes the closed unit disc then there does not exist a continuous ...
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16 views

path connectedness of space of almost commuting matrices

Let $R$ be a topological ring which is a domain. Let $n$ be an integer and let $\zeta_n$ be a $n$-th root of unity. Denote by $X$ the set of $m$ by $m$ invertible matrices with coefficients in $R$ ...
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105 views

Line with two origins is a manifold but not Hausdorff

The line with two origins is $(\mathbb{R} \times \{0,1\})/\sim$ where $(x,0)\sim(x,1)$ for $x\neq 0$. I can see that it is not Hausdorff, since we cannot separate the points $(0,0)$ and $(0,1)$. ...
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93 views

Problem 30 in the Exercises following Chapter 2 in Baby Rudin: How to immitate the proof of Theorem 2.43?

Here's problem 30, the very last one, in the Exercises after Chapter 2 in Walter Rudin's Principles of Mathematical Analysis, 3rd edition: Imitate the proof of Theorem 2.43 to obtain the following ...
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46 views

Condition ici=ic on a topological space is equivalent to if each dense set has dense interior in the space.

I am required to prove the following: Let $(X,\tau)$ be a topological space.Then each dense set has dense interior iff $ici=ic$ holds where $i$ is the interior operator and $c$ is the closure ...
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32 views

Proof of Triangulation Theorem for 1-Manifolds

While I am reading "Introduction to Topological Manifolds" by John M. Lee, I come to see the following paragraph in the proof of Theorem 5.10 pp. 102. Note that Int$\ e\cap\ $Int$\ e'$ is open in ...
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Is a countable, nowhere compact, zero-dimensional, dense in itself, Hausdorff space which is 2nd countable; homeomorphic to space of rationals?

Let $X$ be a countable, nowhere compact, zero-dimensional, dense in itself, Hausdorff space which is 2nd countable. Is $X$ homeomorphic to the space of rationals? $X$ is called nowhere compact when ...
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31 views

Is $S(\mathbb{R}^{d})$ dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$?

Let $S(\mathbb{R}^{d})$ denote the class of Schwartz functions in $\mathbb{R}^{d}$. Is it true that $S(\mathbb{R}^{d})$ is dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$, the locally integrable ...
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30 views

$A$ and $A+y$ are homeomorphic where $A$ is open set

Actually I need to understand $A+B$ is open whenever $A,B$ open set in $\mathbb{R}$ First I want to prove $A$ and translation of $A$ by $y,y\in B$ are homeomorphic $f:A\to A+y, f(x)=x+y$ may be the ...
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24 views

Singleton sets and net criteria for closeness

Theorem. Let $(X,U)$ be a topological space and let $A$ be a subset of $X$. Then $x \in cl(A)$ if and only if there is a net in $A$ that converges to $x$. My question? Does this theorem imply that ...
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38 views

Is this statement true?(covering map)

Let $C,X$ be topological spaces. Let $p:C\rightarrow X$ be a continuous function. Let $U$ be an evenly covered open subset of $X$. Let $V$ be an open subset of $C$ such that ...
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33 views

Showing $\phi(f \cdot g) = \phi(f) + \phi(g)$

For $\phi \in C^1(X; G)$ a cocycle being thought of as a function from paths in X to G, I want to show: $\phi(f \cdot g) = \phi(f) \cdot \phi(g)$. What I'm not sure is how I'm supposed to relate a ...
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39 views

Does there exist a continuous function between the following sets:

Does there exist a continuous function between the following sets: $A.f:(-1,1)\rightarrow (-1,1]$ which is onto and one-one $B.f:\{(x,y):y^2=4x\}\rightarrow \mathbb R$ which is one-one What ...
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62 views

Is the Zariski Topology

if $ K $ is an algebraically closed field, asks: Is there a point $ "w" $ of $ K ^ n $, is closed in the Zariski toplogy?
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48 views

Understanding the mechanics of P-adic topologies

I am trying to work out how it is that we actually work open sets on a p-adic topological space and how I would relate it to open sets in a point set topology. According wiki here: We have that open ...
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58 views

Lie group quotient structure

Let $G$ be a Lie group and $H$ a normal finite subgroup. Let $\pi : G \to G/H$ be the quotient surjection. How would one show that $G/H$ is a Lie group?
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34 views

Prove this map is continuous

$(rcos(t),rsin(t))↦((1/r).cos(t),(1/r).sin(t)), 0≤t≤2pi $ first for $0<r<1$, then for $r>1$ My idea is to say $(rcos(t),rsin(t)) = r .(cos(t),sin(t))$ then the cos and sin map with an ...
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Geometrical Explanation of Borsuk Theorem

Assume $K$, $L$ are $n$-pseudomanifold, and $K$ is compact. Let $f$ be a simplicial map between $K$ and $L$. We denote $n$-simplexes of $K$ and $L$ by $S_n(K)$, $S_n(L)$. Define ...
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22 views

Topology with equivalence of convergence of nets and almost everywhere convergence

I want to show that there is no topology for the set of Lebesgue measurable functions such that the net $<f_n> \to f$ iff $f_n \to f$ almost everywhere. Assume that there exists such a ...
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29 views

Helping me my study of introduction to analysis

I am a math major student who started study math now In my university class , my professor proposed me a few question and I thoought several hours but I can`t write logically so i ask about question ...
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35 views

When does a homogeneous space define a fibration?

Let $G$ be a locally compact and $\sigma$-compact group acting continuously and transitively on locally compact Hausdorff $X$. Then if $x_0 \in X$ and $H_{x_0}$ denotes the isotropy group at $x_0$ we ...
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47 views

Uniqueness of the universal covering space (up to an isomorphism)

Let $Y_1$, $Y_2$ be universal covering spaces of some topological space $X$. I want to show that $Y_1$ are $Y_2$ are isomorphic. Denote $p_1 \colon Y_1 \to X$, $p_2 \colon Y_2 \to X$ the projections. ...
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37 views

Existence of covering space

I would like to know that if $X$ is a connected topological space, there is always a covering space of it, i.e., a continuous map $p:X'\to X$ with the known property.
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37 views

Union over disjoint union

How does the normal union behave over the disjoint union? For instance, if i have some indexed collection of disjoint unions between two sets, what is the union over the whole collection?
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47 views

Continuous Strong-Strong Implies Continuous Weak-Weak

Let $X$ and $Y$ be two Banach spaces and let $T$ be a linear map between $X$ and $Y$. Show that $T$ is continuous strong-strong if and only if $T$ is continuous weak-weak. I can see that $T$ ...
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46 views

Proving an attractor (i.e set with self similarity) is connected

Let $K$ be an attractor for iterating function system of two similarity maps i.e $$K=f_1(K)\cup f_2(K)$$ A similarity map is defined to be $f_i:\mathbb{R}^d\to \mathbb{R}^d$ s.t $$\forall x,y\in ...