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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4
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3answers
157 views

Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...
4
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2answers
226 views

If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable

I need to show that: If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable. I have already showed that every locally compact Hausdorff space ...
4
votes
1answer
90 views

How to make an order isomorphism

Two linear orders $A$ and $B$ have starting points $a_0$ and $b_0$, and have cofinalities $\omega_1$. Let $(a_\alpha )_{\alpha<\omega_1}$ and $(b_\alpha )_{\alpha<\omega_1}$ be cofinal ...
4
votes
2answers
119 views

(Certain) colimit and product in category of topological spaces

Consider the diagram $$(*):\;\;\;X_0 \stackrel{i_0}\hookrightarrow X_1 \stackrel{i_1}\hookrightarrow X_2 \stackrel{i_2}\hookrightarrow \cdots $$ in category of topological spaces. Denote $I$ the ...
4
votes
1answer
97 views

Equal boundaries

I am given that $A \subset B \subset \mathbb{R}$, $A$ is open, $B$ is closed, and that $\partial A = \partial B$. Can I prove from this that $B$ is either equal to the closure of $A$ or it is ...
4
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3answers
112 views

Limits of Topological Vector Spaces

Let $X, Y_1, Y_2, \cdots$ be a sequence of topological vector spaces, and let $f_n : X \to Y_n$ be a sequence of continuous linear maps. Define the product space $\mathcal Y_N := Y_1 \times \cdots ...
4
votes
1answer
69 views

Homeomorphism class of GL_n?

For example, it is easy to see that $GL_1(\mathbb{C})$ is a plane minus a point, and $GL_2(\mathbb{R})$ is $\mathbb{R}^4$ with a topological (half-open) cube removed (since the matrices of determinant ...
4
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3answers
247 views

Prove or disprove that if $f$ is continuous function and $A$ is closed, then $\,f[A]$ is closed.

If $f:X\to Y$ is a continuous function and $A\subset X$ is a closed set, then $f[A]$ is also closed. I know that if $f[A]$ is closed implies $A$ is closed then $f$ is continuous, but I'm not sure ...
4
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3answers
360 views

Proof help. Core-compactness, Hausdorff, Locally Compact

While reading about topologies on continuous function spaces, I've seen remarks that core-compact and locally compact are equivalent for Hausdorff spaces. Now I can clearly see that locally compact ...
4
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5answers
149 views

Nets and Convergence: Why directed indices?

Please do read carefully (I know Nets-Topology-Filters and their interrelations!!!) 1.) Why do we require nets to be indexed by directed sets (apart from it simply works compared to filters and ...
4
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2answers
267 views

Show that the cone of the open interval (0, 1) can not be embedded in any Euclidean space

I've been trying to tackle this problem for some while now, but don't know how to start correctly. I know that the cone on $(0,1)$ is given by $$\text{Cone}((0,1)) = (0,1) \times ...
4
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3answers
186 views

Show that $[0, 1)\times[0, 1)$ is homeomorphic to $[0, 1]\times[0, 1)$ but not to $[0, 1]\times[0, 1]$.

Show that $[0, 1)\times[0, 1)$ is homeomorphic to $[0, 1]\times[0, 1)$ but not to $[0, 1]\times[0, 1]$. When I sketch these spaces it this statement makes sense intuitively because $[0, 1]\times[0, ...
4
votes
1answer
170 views

What do we mean when we say a differential form “descends to the quotient”?

Let $S$ be a surface and let $f:S\rightarrow S$ be a diffeomorphism. We define the mapping torus $M_f$ of the pair $(S,f)$ to be the quotient $$(S\times I) /\sim \quad \text{ where } \ (1,x) \sim ...
4
votes
1answer
319 views

Why does the (topology given by) Hausdorff metric depend only on the topology?

If I have a compact metric space $(X,d)$, I can define the Hausdorff metric on the set $K(X)$ of all non-empty compact (equivalently, closed) subsets of $X$ as $$d_H(A,B) = \max ( \sup_{x \in A} ...
4
votes
1answer
415 views

Separately continuous functions that are discontinuous at every point

What are some good examples of separately continuous functions $f: X \times Y \rightarrow Z$ that are discontinuous at every point? Here's a theorem to rule out some spaces: link for a reference ...
4
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3answers
486 views

Prove that $Y$ is Hausdorff iff $X$ is Hausdorff and $A$ is a closed subset of $X$

Let $X$ be a topological space and $A$ a subset of $X$. On $X\times\{0,1\}$ define the partition composed of the pairs $\{(a,0),(a,1)\}$ for $a\in A$, and of the singletons $\{(x,i)\}$ if $x\in ...
4
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2answers
279 views

When a covering map is finite and connected, there exists a loop none of whose lifts is a loop.

I've read the following exercise. Let $p:\tilde X\to X$ be finite connected covering map. Show that there exists a loop in $X$ none of whose lifts is a loop. I can't understand why it's supposed ...
4
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2answers
226 views

Tietze extension theorem for complex valued functions

Why is this theorem always only stated for real valued functions, and not for complex valued functions? Thanks.
4
votes
1answer
281 views

Relation between convergence class and convergence space

A convergence class is defined from nlab A convergence space is a set $S$ together with a relation $→$ from $ℱS$ to $S$, where $ℱS$ is the set of filters on $S$; if $F→x$, we say that $F$ ...
4
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2answers
940 views

If $f,g:X \to Y$ are continuous and $Y$ is $T_2$, then $\{x \in X\,|\,f(x)=g(x)\}$ is closed

I'd like to know if the following proof is valid. The only thing I'm not sure about (though I can't see why it's invalid if it is) is if we can always use the Hausdorfness of $Y$ to separate an open ...
4
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2answers
386 views

Compact subspaces of the Poset

On page 172, James Munkres' textbook Topology(2ed), there is a theorem about compact subspaces of the real line: Let $X$ be a simply-ordered set having the least upper bound property. In the order ...
4
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1answer
608 views

If $X$ is infinite dimensional, all open sets in the $\sigma(X,X^{\ast})$ topology are unbounded.

As in the title, if $X$ is infinite dimensional, all open sets in the $\sigma(X,X^{\ast})$ topology are unbounded. The $\sigma(X,X^{\ast})$ topology is the weakest topology that makes linear ...
4
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1answer
703 views

What is the generating set of the Vietoris topology?

Here is what Kechris says about the Vietoris topology, Let X be a top. space. We denote by K(X) the space of all compact subsets of X equipped with the Vietoris topology, i.e., the one generated by ...
4
votes
2answers
570 views

which of the following metric spaces are separable?

which of the following metric spaces are separable? $C[0,1]$ with usual 'sup norm' metric. the space $l_1$ of all absolutely convergent real sequences, with the metric ...
4
votes
3answers
245 views

Prove one set is a convex hull of another set

Define two sets: $A = \{x \in \{0,1\}^n : \lVert x \rVert_1 \leq k\}$ is a finite set of binary vectors; $B = \{x \in [0,1]^n : \lVert x \rVert_1 \leq k\}$ is an infinite set of real-valued vectors, ...
4
votes
2answers
956 views

Definition of metrizable topological space

I am learning a bit about Topology through independent study. I am using Bert Mendelson's "Introduction to Topology - 3rd Edition". I have a question on one of the book's example and related ...
4
votes
2answers
2k views

Cantor set: Lebesgue measure and uncountability

I have to prove two things. First is that the Cantor set has a lebesgue measure of 0. If we regard the supersets $C_n$, where $C_0 = [0,1]$, $C_1 = [0,\frac{1}{3}] \cup [\frac{2}{3},1]$ and so on. ...
4
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1answer
382 views

Dealing with connectness and compactness of matrices.

Consider the set of all $n\times n$ matrices with real entries, considered as the space $\mathbb{R^{n^2}}$ What can we say about connectedness and compactness of the following sets? The set of all ...
4
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2answers
108 views

Spaces admitting each singleton as zero set

I'm looking for classes of spaces $X$ having the property that for each $x_0 \in X$ there is a continuous map $f:X \to \mathbb R$ such that $Z(f) := f^{-1}(0) = \lbrace x_0\rbrace$. Examples are: ...
4
votes
1answer
1k views

Definition of Basis for the Neighborhood System

I'm trying to learn a bit about topology through independent study. I've been using Bert Mendelson's "Introduction to Topology - 3rd edition". I'm having a lot of fun but I'm a bit confused regarding ...
4
votes
2answers
956 views

Is the complement of a countable set in $\mathbb{R}$ dense? Application to convergence of probability distribution functions.

I am wondering if we have a set $A\in\mathbb{R}$ that is countable, whether $A^{c}$ is dense in $\mathbb{R}$? I thought I saw this quoted somewhere on google but I can not find it again! I am working ...
4
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2answers
500 views

Why is this not the Stone-Čech compactification?

I've been reviewing properties of compactifications, and I came across this question. If $cX$ is a compact metric space and $X$ is a dense subset of $cX$, where $X \neq cX$, why is $cX$ not the ...
4
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2answers
569 views

A concrete example of a choice function

I'm trying to understand a bit what lies behind the Axiom of Choice, and I was wondering, are there concrete examples of a choice function on the Borel set? The Borel set seems nice enough for a ...
4
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5answers
1k views

Complete implies locally compact in length metric space?

I am confused. The way I see it, in a complete metric space, closed balls of finite diameter are compact since they are complete and totally bounded. Consequently a complete metric space is locally ...
3
votes
1answer
33 views

Say $X$ is $T_2$, $f: X \to Y$ is continuous, $D$ is dense in $X$ and $f|_D :D \to f(D)$ is a homeomorphism. Then $f(D) \cap f(X- D) = \emptyset$

I've been looking into the following question: Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism (It is also given that $X$ is ...
3
votes
2answers
60 views

Prove that $f_A (x) = d({\{x}\}, A)$, is continuous.

Prove that: Let $(X, d)$ be a metric space, and let $A$ be a subset of $X$. The function $f_A\colon X\rightarrow \mathbb{R}$, defined by $f_A (x) = d({\{x}\}, A)$, is continuous. Honestly, I ...
3
votes
2answers
80 views

Motivation for separation axioms

I have recently been studying different separation and countability axioms in topology. I am looking for a motivation for why such a refined division of different axioms was made and is studied. I am ...
3
votes
1answer
36 views

Classification of Proper Maps between domains in $\mathbb{R}^n$

Suppose $f:D_1\to D_2$ is a continuous map between domains in $\mathbb{R}^n$. Show that $f$ is proper iff for every sequence $(x_n)$ in $D_1$ which accumulates only on $\partial D_1\cup\{\infty\}$, ...
3
votes
2answers
74 views

Why do we need surjectivity in this theorem?

In class we proved the following theorem: Given $X_1,X_2$ ordered sets. Then any surjective increasing $\phi: X_1 \to X_2$ is continuous wrt the interval topology on $X_1$ and $X_2$. I was asked to ...
3
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1answer
83 views

Approximation of a strongly measurable function by a sequence of simple functions.

Let $(X, \mathcal{A})$ be a measurable space and let $E$ be a normed space. $(i)$ $f:X \rightarrow E$ is called Borel measurable if $f^{-1}(B) \in \mathcal A$ for all $B \in \mathcal{Bo}(E)$ where ...
3
votes
1answer
108 views

Are $L^\infty$ bounded functions closed in $L^2$?

Is the set $\{ m \in L^2(0,1) : |m|_{L^\infty}\leq A \}$, (i.e. the set of $L^2$ functions with bounded $L^\infty$ norm) a closed subset of $L^2$? (Closed in the topology induced by the $L^2$-norm)
3
votes
1answer
99 views

Prove existence of disjoint open sets containing disjoint closed sets in a topology induced by a metric.

Question: Let $(X, d)$ be a metric space. Let $A$ and $B$ be disjoint subsets of $X$ that are closed in the topology induced by $d$. Prove that there exist disjoint open sets $U$ and $V$ such that ...
3
votes
1answer
83 views

“isometric invariant” vs “isometric” what do these term mean?

I am now hopelessly confused: There is Hilberts Theorem https://en.wikipedia.org/wiki/Hilbert%27s_theorem_%28differential_geometry%29 . that implies that there are no isometric embeddings of the ...
3
votes
1answer
99 views

Build a topological manifold starting from a set.

Suppose you are given a generic set $X$. There exist sufficient and non-trivial conditions that ensure the existence of a topology $\tau_X$ on X such that the topological space $(X,\tau_X)$ is a ...
3
votes
1answer
131 views

Proof that the cardinality of continuous functions on $\mathbb{R}$ is equal to the cardinality of $\mathbb{R}$.

Proof that the cardinality of continuous functions on $\mathbb{R}$ is equal to the cardinality of $\mathbb{R}$. I think is should be proved with the help of Cantor-Bernstein theorem. It is easy to ...
3
votes
2answers
154 views

Reference Request to Prepare for Hatcher's “Algebraic Topology”

Hatcher himself has an excellent and always generously free set of notes on point- set topology: http://www.math.cornell.edu/~hatcher/Top/TopNotes.pdf It includes up to quotient spaces. It seems ...
3
votes
2answers
127 views

Urysohn's lemma with Lipschitz functions

In a complete and separable metric space $(X,\mathrm{d})$ given an open set $U$ and a closed set $K\subset U$. Is it possible to find a Lipschitz function $f$ such that $f|_K=1$ and $f|_{X\setminus ...
3
votes
2answers
100 views

Connected topological spaces, product is connected

Show that if $(X_i)_{i \in \mathcal I}$ where $X_i$ is a topological space for every $i \in \mathcal I$, then $X_i$ is connected for every $i$ if and only if $\prod_{i \in \mathcal I} X_i$ is ...
3
votes
2answers
291 views

If two Borel measures coincide on all open sets, are they equal?

Let $X$ be a topological space and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra. That is, $\mathcal{B}(X)$ is the smallest $\sigma$-algebra on $X$ containing all the open sets. Let $\mu, \eta : ...
3
votes
1answer
119 views

Independence of $H(f)=\int_M \alpha \wedge f^* \beta$ on choice of $d\alpha=f^*\beta$?

I came across the following UCLA qual question while studying for my upcoming qual: Let $f: M^{4n-1} \to N^{2n}$ be a smooth map between closed connected oriented manifolds of the indicated ...