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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126 views

Is it true that a set is compact iff it is closed, bounded, and has finite measure?

I'm sure that this holds for $\mathbb{R}^n$ and for $L^p$ spaces. Is it true in general?
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112 views

characterization of compactness

When $X$ is an arbitrary topological space, I need to know which of the followings are true or false $1$. If $X$ is compact, then every sequence in $X$ has a convergent subsequence. $2$. If every ...
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218 views

the geometric boundary of a manifold

Let $X$ be a topological manifold with geometric boundary $\partial_g X$ (here the subscript $g$ to indicate geometric boundary which is a different notion from topological boundary $\partial_t X$). ...
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850 views

How are a set of matrices a topological space?

For instance, is the topology on the set of all $2 \times 2$ real matrices basically $\mathbb{R}^4$
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48 views

“Agreement Domain” of Function Families

Given a family of functions $\{f_i : X \to Y\}_{i \in I}$ between topological spaces $X$ and $Y$, I define an operation $\bigcap\limits^{\scriptscriptstyle\text{dom}}$ on the family such that ...
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99 views

How products in Top and Set are related?

Are product morphisms for a categorical product in Top the same as for categorical product morphisms in Set? More generally: How product morphisms for Top are characterized?
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920 views

Two problems: When a countinuous bijection is a homeomorphism? Possible cardinalities of Hamel bases? [closed]

Let $X$ and $Y$ be topological spaces and let $f : X\rightarrow Y$ be a continuous bijection. Under which of the following conditions will $f$ be a homeomorphism? (a) $X$ and $Y$ are complete metric ...
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1answer
352 views

Constructing a local nested base at a point

I am trying to prove the following: "Let $X$ be a first countable space and $x$ a member of $X$. Prove that there is a local nested basis $\{S_n\}_{n=1}^\infty$ at $x$." Since $X$ is first ...
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1answer
486 views

Closed set in a Hausdorff topological space [duplicate]

Possible Duplicate: $X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed I'm trying to prove: If $X$ is a Hausdorff topological space and $\Delta \subset X\times X$ such ...
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194 views

Two questions about continuity of function between topological spaces

Let $X$ and $Y$ be topological spaces and suppose $f: X \to Y$ is continuous. If $f$ is continuous on $U \subset X$, will the restriction $f_U :U \to Y$ be continuous, if we consider $U$ to be a ...
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1answer
111 views

Example for $\Omega$ so that $\partial\Omega \neq \partial\bar{\Omega}$

Let $\Omega$ be an open subset of $\mathbb R^n$, with $\Omega \neq \emptyset$ and $\Omega \neq \mathbb R^n$. Can you give an example where $\partial\Omega \neq \partial\bar{\Omega}$, and how can one ...
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152 views

If $f:M\to N$, $g:N\to P$ continuous and $g\circ f: M\to P$ are homeomorphism, and $g$ is injective, then $g, f$ both are homeomorphisms

If $f:M\to N$, $g:N\to P$ continuous and $g\circ f: M\to P$ is a homeomorphism. And $g$ is injective (or $f$ is surjective) then $g, f$ both are homeomorphisms. I don't know how to prove it. I ...
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827 views

Regular, but not a normal topological space

Let $X=[0,1)\times[0,1)$, $\tau$ its topology with base $$\beta = \{ [a,b)\times[c,d): 0 \leq a < b \leq 1, 0 \leq c < d \leq 1 \}\;.$$ Please help me prove, that it is regular, but not a ...
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72 views

a question on Lindelöf spaces

Let $X=\{(x,y)\in {\Bbb R}^2:y>0\}$ is the subspace of ${\Bbb R}^2$ with the usual topology, then it is still Lindelöf? If not, with which topology can $X$ be made to be Lindelöf?
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1k views

Complex conjugation continuous function

How can I show that complex conjugation is a continuous function? I tried looking at open sets $U$ and then the preimage. Can assume preimage is not empty so that if $z$ is in preimage then $f(z) \in ...
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1answer
1k views

An open subset $U\subseteq R^n$ is the countable union of increasing compact sets.

Why is this true? I think I can find a countable union of compact sets $\cup_{k=1}^\infty X_k$ such that $\cup X_k \subseteq U$ and the lebesgue measure of $U \setminus \cup X_k$ is zero. (for any ...
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261 views

Homeomorphic spaces : set with two elements

The following is taken from wikipedia: http://en.wikipedia.org/wiki/Finite_topological_space 2 points Let $X = \{a,b\}$ be a set with 2 elements. There are four distinct topologies on ...
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281 views

What is the fundamental group of $\mathbb{CP}^1$ minus a finite set of points?

Let $\mathbb{C}\mathbb{P}^1$ be the projective space. Let $a_1, \ldots, a_n \in \mathbb{C}$. What is the fundamental group $\pi_1(\mathbb{CP}^1\backslash \{a_1, \ldots, a_n\})$?
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120 views

Prove that the sets are open.

I'm having some trouble with the following questions: Let $S$ be any set and $\epsilon$ > 0. Define $T$ = {$t$ $\in$ $\mathbb{R}$ : |$t - s$| < $\epsilon$ for some $s \in S$}. Prove that T is ...
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1answer
837 views

Equivalence of continuity definitions

How to show that $(1)\Longleftrightarrow (2)$ in metric spaces ? pre-image of open sets are open $\delta$-$\epsilon$ definition of continuity
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1answer
359 views

Properties of generalized limits aka nets

I want to find some article or a book which contains all general properties of nets. Of course some of them similar to properties of sequences with almost the same proofs, but I don't fill the edge, ...
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276 views

What is “uniform cover”?

I was reading Wikipedia's definition of uniform spaces in terms of uniform covers. I wonder how a "uniform cover" of a set is defined? I just can't find it anywhere. Thanks!
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140 views

Are all subspaces of the rationals order topologies (with respect to some linear order)? All closed subspaces?

This is a follow-up to this question. Let $Y \subset \mathbb{Q}$, and give $Y$ the subspace topology. Is there necessarily a linear ordering $<$ of $Y$ (possibly different from the order inherited ...
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1k views

What is 'an identification map'?

From Husemöller's 'Fiber Bundles' (slightly rephrased): Proposition: Consider a bundle $\xi: E \to B$, and a mapping $f: B' \to B$. Then for any $s \in \Gamma(\xi)$ there is a $\sigma: B' \to ...
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159 views

quasi-compact and compact in algebraic geometry

In reading Hartshorne,a topological space is quasi-compact if each open cover has a finite subcover(P80).Isn't it the definition for compactness of topological spaces?Am I right?Is quasi-compactness ...
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529 views

Showing that Mg, the Mapping Class Group of the 1-Torus, is $SL(2,\mathbb Z)$

All: I am trying to figure out the mapping class groupof the torus ; more accurately, I am trying to show that it is equal to $SL(2,\mathbb Z)$. The method: every homeomorphism h: $ T^2 \rightarrow ...
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1answer
292 views

property of Euler characteristic

I read the statement that the Euler characteristic is always additive with respect to closed-closed union, which means that $\chi(X\sqcup Y) = \chi(X)+\chi(Y)$ if $X$ and $Y$ are closed. And I read ...
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1answer
238 views

Is the space of positive (semi- or not) definite correlation matrices Polish?

Title basically says it all: Is the space of positive (semi- or not) definite correlation matrices Polish? As an aside, I'm interested in general comments/references about the space(s). Edit: For ...
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1answer
478 views

Discrete and metric topologies equivalence

Given a set $X$, define a function $d:X\times X\rightarrow \mathbb{R}$ by $d(x,y) = 1$ if $x\neq y$ and $d(x,y)=0$ if $x=y$. Show that the metric topology on $X$ is equal to the discrete topology.
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389 views

$L^{\infty}$($E$) could be separable for a measurable set $E$

I know that in general, $L^{\infty}$($E$) is not separable, like for example, if $E$ = [$a$,$b$]. But wouldn't $L^{\infty}$($E$) be separable if $E$ = $\mathbb{Q}$, i.e. the set of rational numbers? ...
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114 views

If $X\times Y\rightarrow Z$ is continuous then the partial maps $\langle\cdot , y_{0}\rangle$ and $\langle x_{0}, \cdot\rangle$ are continuous

Let $f:X\times Y\rightarrow Z$ be a mapping of topological spaces, the product given the product topology. Then for all $y_{0}\in Y$, the map $f(.,y_{0}):X\rightarrow Z$ is continuous. Any hints on ...
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132 views

how can i show that £ is a topology on a set X?

Consider the set X={a,b}, and the collection £ given by £={ Ø, {a},{b},X }. show that £ is a topology on X. I know that from the definition of topological space i must consider to show ...
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151 views

If $f(z)=(g(z),h(z))$ is continuous then $g$ and $h$ are as well

If $f(z) = (g(z),h(z))$ is continuous then $g$ and $h$ are as well. The converse is easy for me to prove, but I'm not seeing how to prove it using the terminology of open sets and not metric ...
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1k views

Show that a subset $A$ of $R$ is open iff it is countable union of open intervals

Let me prove subset $A$ of $\mathbb{R}$ is open if and only if it is a countable union of open intervals. For all $x \in A$ there is an $\epsilon > 0$ such that $(x-\epsilon,x+\epsilon)$ is an ...
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22 views

$M$ is open in $Y$ and $M$ is open in $Z$ then $M$ is open in $X$

Is it true? $X= Y \cup Z$, $M$ is a subset of $Y \cap Z$. Suppose that $M$ is open in $Y$ and $M$ is open in $Z$ then $M$ is open in $X$.
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68 views

Generalizing limits of sums, products, and quotients of sequences to abstract topological spaces?

Introductory real analysis books usually state some properties about limits of sequences. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then: $\lim (x_n+y_n) = x+y$ $\lim c x_n = cx$, for $c \in ...
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1answer
27 views

Is it true that factor spaces are T4 if product space is T4?

I use the following definition of $T_4$-space: for any two disjoint closed sets $A$, $B$ there exist disjoint open sets $U$, $V$ containing $A$ and $B$ respectively. Is it true that factor spaces are ...
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57 views

How does convergence imply continuity?

I'm trying to develop some background understanding to eventually prove the following: ....................... Let $M$ and $N$ be metric spaces and let $f : M \rightarrow N$ be a map. Show that $f$ is ...
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47 views

Are these subsets open, closed, both or neither?

I'm teaching myself topology using a text I found online. Right now I'm reviewing "Metrics." Please let me know if my answers are correct, and If my reasoning is accurate and complete. I think (c)and ...
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30 views

Does finite covering dimension imply local compactness?

I have a space which is not locally compact and I'm trying to see if I can say anything about the dimension of the space. I suspect that it is not finite dimensional but I have thus far been unable to ...
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1answer
32 views

Why does the Hausdorff metric need to be defined on bounded subsets only?

Claim: Suppose $X$ is a non-empty set and $d$ is a metric on $X$. Let $S(X)$ denote the collection of all non-empty closed bounded subsets of $X$. For each $A$ and $B$ in $S(X)$, define $$h(A,B) = ...
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56 views

What is a topology minus the axiom that $\varnothing \in \tau$.

For instance, this is the case with defining $U \subset \Bbb{N}$ to be open iff $\sum_{x \notin U} \frac{1}{x} \lt \infty$ if we let $\sum_{x \notin \Bbb{N}} = 0$. I can't seem to get $\varnothing$ ...
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55 views

If a line bundle admits a non-vanishing section then it is trivial

Suppose $\pi:E\to B$ is a line bundle. Let $s:B\to E$ a non-vanishing section, i.e. for every $b\in B$ $s(b)\ne 0$ and $\pi\circ s=Id_B$. I have to prove that the line bundle above is trivial. Idea: ...
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70 views

Problem with the proof 0f “ the intersection of closed sets is closed”.

I have been reading this text http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF, and before I address my main question, I want to note that the author, in the same section, ...
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25 views

denseness of polynomials in bounded borel measurable functions

Let $K\subseteq \mathbb{R}$ be compact, consider $B(K)$ the set of all bounded borel measurable functions $f:K\to \mathbb{C}$ and endow $B(K)$ with the uniform norm, so you obtain a Banach space. My ...
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68 views

What is the homeomorphism between a disk and an ellipse?

A disk/circle is defined by $$C = \{(x,y) \in \mathbb{R^2} : x^2 + y^2 \leq r^2\}$$ An ellipse is defined by $$E = \{(x,y) \in \mathbb{R^2}: x^2/a^2 + y^2/b^2 \leq 1 \}$$ How can we define a ...
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71 views

Is simply connectedness is a topological property?

A topological space $X$ is called simply-connected if it is path-connected and any continuous map $f:S^{1} \to X$ (where $S^1$ denotes the unit circle in Euclidean 2-space) can be contracted to a ...
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50 views

Every quasi-compact scheme has a closed point

I know this question has been asked here before, but I have trouble understanding the following proof, taken from a Schwede's write-up. I have underlined the bit I don't understand. In particular, ...
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134 views

a good modern topology book

I want to study an advanced modern book on topology, but I couldn't find any. I've already studied the first chapters of Munkres' book, but it is not as advanced as books such as Engelking's topology, ...
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50 views

False proof that all topological groups are discrete: what went wrong?

I can't seem to find the mistake in this obviously false proof I've thought up while trying to understand topological groups. It's pretends to prove the discreteness of all topological groups. Let ...