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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Showing that an identity map for a metric space in $\mathbb{R}^2$ is continuous but that its inverse isn't.

Suppose that $A=(\mathbb{R}^2,d)$ is a metric space with $d(x,y)=||x-y||$. I would like to show that if I have an identity map from $I:A \to \mathbb{R}^2$ with its euclidean distance function, then ...
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46 views

point set topology: closed points dense

Let $X$ be an irreducible finite dimensional Jacobson scheme (i.e. the closed points lie dense and the underlying topological space is sober). If one chooses for every closed point $x \in |X|$ an open ...
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32 views

Separability of a certain space of continuous functions

Let $I$ be a separable, locally compact Hausdorff space, and let $V$ be a separable, locally convex, complete topological vector space. Consider the function space $C(I, V)$ with the compact-open ...
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31 views

Products of cells in a CW complex

Suppose I have a manifold which has a CW structure with cells $e^0 \cup e^1 \cup e^2$, where $e^i$ represents an $i$-cell. If I took the direct product of this manifold with another manifold which has ...
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32 views

Definition of Jets

Can someone help me with a definition of jets between Cr manifolds. I want to avoid using inverse limit at infinity, but how do we define jets then ?
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16 views

A question in the Maximum and Minimum Value Theorem

This is an excerpt from Bartle's The Elements of Real Analysis. I'm having trouble trying to understand the second sentence of the proof. What does $f(x_n)\ge M-\frac{1}{n}$ show exactly?
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50 views

Homeomorphisms on Zariski topologies

I'm looking for a continuous bijection from a compact space to a non-Hausdorff topological space which isn't a homeomorphism. Since the identity $f:\mathbb{Z}\rightarrow\mathbb{Z},\ x\rightarrow x$ is ...
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45 views

What is this space homeomorphic/homotopy equivalent to?

Let $X \subset \mathbb{C}^2$ be given by the equation $$|z|^2 + |w|^2=1$$ and let $A \subset X$ be given by $|z|=1,\ w=0$. This question requires me to find the relative homology groups $H_n(X,A)$, ...
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27 views

about a topological group

I can't solve this exercise from topology Munkres page 172: Let $G$ be a topological group. (a) Let $A$ and $B$ be subspaces of $G$. If $A$ is closed and $B$ is compact, show that $A\cdot B$ is ...
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45 views

two disjoint closed sets in a locally compact Hausdorff space

Let $X$ be a locally compact Hausdorff space and $C_{0}(X)$ be the $C^{*}$ algebra of all continuous complex functions on $X$ which vanish at infinity. For a closed subset $F$ of $X$, denote by ...
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26 views

Finding a uniformly continuous function from a non complete space to $\mathbb{R}^{+}$

Assume that $(X, d)$ is not complete. Prove that there exists a uniformly continuous function $f:X \rightarrow \mathbb{R}^{+}$ such that $\inf_{X} f(x) = 0$. We are guaranteed the existence of a ...
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47 views

Showing $\mathbb{R}$ is a completion of $(\mathbb{Q}, | \cdot |)$

Recall the definition of a completion: A complete metric space $(Y, d)$ is said to be a completion of another metric space $(X, d)$ if there exists a map $f: X \rightarrow Y$ such that $f$ is an ...
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41 views

A lemma on function spaces

This is a lemma about function spaces. I'm not really understanding it however. Can someone try explaining it to me? Lemma: let $X$ be in |SET| $(Y, d)$ in |MET|, $f_n$, $f$ is in $Y^X$. Then $f_n\to ...
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65 views

Connected space whose every subspace is disconnected

We know that a subspace of a connected space can be disconnected eg. $\mathbf{Q} \in \mathbf{R}$ where $\mathbf{R}$ is connected but $\mathbf{Q}$ is totally disconnected as a subspace. My question ...
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65 views

$\mathbb{R}^n$ $\backslash$ $\mathbb{R}^k$ what does this mean?

$\mathbb{R}^n$ $\backslash$ $\mathbb{R}^k$ I saw this in my topology assignment. The question was about quotient spaces and homeomorphisms. I have never seen this expression before so it doesn't ...
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30 views

$L \cap V = \overline{L} \cap V \implies L$ open in $\overline{L}$. [Solved]

how do I prove this? My attempt: Let $x \in L$, and for some open set $V \ni x$ suppose that $L \cap V = \overline{L}\cap V$, also maybe use the fact that $L \cap V$ is closed in $V$. If $x \in L$ ...
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96 views

Show that $f(\bar A) \implies \overline{f(A)}$.

Def) X:a metric space, $Y\subset X$: a subset. A point $x\in X$ is adherent to Y if $B(x;r) \cap Y \neq \emptyset \quad \forall r > 0.$ Def) $\bar Y := \{x\in X \mid x \text{ is adherent to } Y\}$ ...
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48 views

Is $(\mathbb{R},+)$ a smooth manifold?

I feel like $(\mathbb{R},+)$ is, but I'm not really sure. How would I know whether or not it is?
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43 views

Examples and counterexamples in dimension theory

I am looking for examples of topological spaces that are interesting from a dimension theoretical point of view - in particular I am looking for these three notions of dimension: Little inductive ...
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33 views

fundamental group of $\mathbb{C^*}/\{e,a\}$

I'm taking an intro to topology course, and am having trouble with this question. What is the fundamental group of $\mathbb{C^*}/\{e,a\}$, where $e$ is the identity homomorphism and $az=\overline{z}$. ...
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35 views

upper hemicontinuity

Let $g: \mathbb R^2_+ \to \mathbb R_+$ and $h: \mathbb R^2_+ \to \mathbb R_+$ continous functions. For every $ t \in \mathbb R_+$, 1) $g(t, \cdot)$ has a unique maximum at $V(t)$ where $V: \mathbb ...
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23 views

How does topological dense subgroup induces properties in the larger group?

Let G denote the group of orientation-preserving isometries of the plane; equivalently, the group of affine transformations of the complex field C of the form $z \rightarrow \alpha z + \beta$ ...
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61 views

Why is the vertex called non-manifold vertex?

I am working on triangle meshes in one 3D reconstruction project for a while. I know what one manifold vertex looks like and how to detect them. But I hope to understand the definition of non-manifold ...
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14 views

Calabi homomorphism of the disk

There is a fact that the homomorphism $Diff_0^{\infty}(\mathbb{D},\partial\mathbb{D},area)\to \mathbb{R}$ is surjective, we can use Calabi homomorphism to prove it, where ...
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24 views

What is the classification theorem of simple Lie groups?

I've seen this thrown around a bit, but I can't find what the theorem actually states? Can anyone help?
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53 views

Examples of topologies between norm and weak star

Let $X$ be a normed vector space and $X^\ast$ denote its continuous dual. The norm on $X^\ast$ is given by $\|\varphi\|=\sup_{\|x\|=1}|\varphi(x)|$. The weak star topology on $X^\ast$ is the weakest ...
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34 views

Question about singular homology

in order to prove that $H_0(X)\simeq \mathbb{F}$, $\mathbb{F}$ is the unitary commutative ring we have to prove that $C_0(X)/B_0(X)\simeq \mathbb{F}$ since we have that $C_0(X)$ is generated by the ...
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24 views

Non-meager set of baire property contains perfect subset

I am in need with some help to understanding a proof. Here is the statement and proof. Let B be the collection of all sets of the baire property in R and M be the meager sets in R. STATEMENT : Let ...
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17 views

What's an example of a Quotient/Identification Space for this topological space?

I haven't been able to find an example with numbers anywhere on the internet and was hoping someone could help. If I have $X = \{1, 2, 3\}, \tau = \{\emptyset, \{1\}, \{2\}, \{1, 2\}, X\}$ as my ...
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97 views

understand proof of compactness in product topology

I am trying to understand the following reasoning. Call $\mathcal{F_\lambda}$ the set of functions $a:\mathbb{N} \to \mathbb{R}$ for which $Na(i) := \sum_{j \in \mathbb{N}} n_{ij} a(j)\leq \lambda ...
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15 views

How to use a base to prove something is sequentially compact.

I know this is not very specific but I'm studying for a topology exam and this is one of the things I need to know how to do. I know that part of the process is showing it converges. I was hoping ...
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42 views

Existence of bijective function

Does there exist a bijective map $f$ from $\mathbb R^2$ to $\mathbb R^3$ such that $f$ and $f^{-1}$ are both differentiable ? My answer was that since $\mathbb R^2$ and $\mathbb R^3$ are not ...
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24 views

History of vectorial bundles in articles or papers?

I'm looking for an article or book that gives a thorough and interesting history of bundles and vectorial bundles in algebraic topology. I'm looking for it for my own learning, please help its ...
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32 views

Antipodal map and parallel transport on $S^3$

I'm reading Thurston and Levy's "Three-dimensional geometry and topology" where they have an informal explanation of what life in $S^3$ would look like. For the following detail that I am concerned ...
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29 views

A question about open “balls”

I've been recently learning Topology and I'm struggling to visualize open balls. For instance, on $\mathbb{R}^2$ and $\mathbb{R}^3$ given a metric like say $d_\infty(x,y)=\sup\{|x_1-y_1|,|x_1-y_2|\}$ ...
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33 views

Hairy ball theorem, projections and L.I. vectors

I'm trying to understand this paper which proves that not every unimodular row is completable by invertible matrices: Why we have these implications: There are two linearly independent vectors at ...
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28 views

Some fundamental relations in topology

Are the following relations correct? $\ \{ Normed\, Vector\, Spaces\} \subset \{Topological\, Vector\, Spaces\} \subset \{Uniform \,Spaces\} \subset \{Topological\, Spaces\}$ Then $\ \{Normed\, ...
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25 views

An example of a Lindelöf topological space which is not $\sigma$-compact

I am looking for an example of a Lindelöf topological space which is not $\sigma$-compact. I have looked in Counterexamples in Topology, but, if I am not wrong, all the examples there which meet my ...
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22 views

Convex in $ \mathbb{R^n}$

Prove that: [A be a convexe part $(A\subseteq \mathbb{R^n})] \implies [\forall x_1,x_2,...x_n\in A ,\forall\alpha_1,\alpha_2,...\alpha_n\ge0 $ $with$ $ \ \alpha_1+\alpha_2+...+\alpha_n=1 ...
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15 views

Constructing a smoothly varying basis without singularities

I am trying to construct a smoothly varying and a differentiable basis to map a vector in $\mathbf{B}:\mathbb{R}^3 \to \mathbb{R}^3$. Given a vector field $\mathbf{n}(\mathbf{x})$ where $\mathbf{n} = ...
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25 views

Removing non-isolated fixed points

Is it true that if $f$ is a homeomorphism of ${\Bbb R}^n$, then there are other homeomorphisms $g$ of ${\Bbb R}^n$ arbitrarily close to $f$ (in the compact-open topology) such that every fixed point ...
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65 views

How I can prove euler characteristic of this complex is zero

$P$ is the poset of all nonempty subsets of $\{ 1, 2, 3, ....,n\}$ under set inclusion. Show that reduced euler characteristic of $\Delta P$ = $ 0$ I tried induction... but failed.
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How do i show that the Riemann sphere and the one-point compactification of $\mathbb{C}$ are homeomorphic?

Let $\mathbb{C}\cup\{\infty\}$ be the one-point compactification of $\mathbb{C}$. Let $S^2$ denote the 2-sphere in $\mathbb{R}^3$. Defne $\zeta:S^2\rightarrow \mathbb{C}\cup\{\infty\}$ as ...
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30 views

Contractions and Fixed Points

I'm working on a question in Munkres: If $f$ is a contraction and $X$ is compact, show $f$ has a unique fixed point. Here's my attempt at a solution so far. $f$ is continuous, choose $\epsilon = ...
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43 views

A special filter on cartesian product of sets

The following is inspired by this article in nLab (in attempt to simplify it using my notions of funcoids and reloids, which notation is however outside of the scope of this question). Fix a set $U$. ...
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42 views

How to deform a curve in specific manner

I am wondering whether we can deform a path in specific ways continuously i mean if there is a closed piece wise $C^1$ smooth path which has to be deformed to another piece wise $C^1$ smooth path. Let ...
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27 views

Cluster Points of a Convergence Space

I'm trying to find the characteristic properties (axioms) of cluster points in a convergence space. I've come up with a minimal two: (let $\mathrm{adh}(\mathcal{F})$ be the set of cluster points of ...
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21 views

What can we say about the space just by looking at its Borel sets?

What can we say about a compact space $X$ just by looking at the Borel sets of $X$? In general, it seems that not much but maybe it is still not a bad question. For instance, let $X$ be a compact ...
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24 views

$(0,1)^\omega$ homeomorohic to $R^\omega$?

Since $(0,1)$ is homeomorphic to $R$ and an infinite product of homeomorphisms is a homeomorphisn?
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50 views

topological vector space of measure functions

Let $(X, \mathcal X, \mu )$ be a measure space, and let $ L(X)$ be the space of measurable functions $f: X \to \mathbb C$. Show that the sets $B(f, \epsilon ,r ): = \{ g \in L(X) : \mu( \{ x : | f(x) ...