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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Unions and intersections of compact subsets

I'd really appreciate some input on these two proofs regarding unions and intersections of compact subsets (under additional necessary conditions). Namely, are my proofs valid? Could they be ...
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239 views

where can i find this A.H. Stone's theorem proof?

can someone tell me where can i find a proof of the following theorem (by A.H.Stone) : "an uncountable product of Hausdorff non-compact spaces is never normal " ? thanks in advance !
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223 views

Characterization of the Closure of a Set

Say I have a metric space $(X, d)$ and a set $A\in X$. I want to prove that if $a \in \overline{A}$ then there is a convergent sequence $\{x_n\}$ that converges to $a$. Could I have any help?
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137 views

is the converse true: in a simply connected domain every harmonic function has its conjugate

The question is. Is the converse true: In a simply connected domain every harmonic function has its conjugate? I am not able to get an example to disprove the statement.
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290 views

Totally ordered set with greater cardinality than the continuum

Does there exist a totally ordered set $S$ with cardinality greater than that of the real numbers? Sequences are continuous functions with domain $\mathbb{N}$ and paths are continuous functions with ...
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120 views

Segment of $\mathbb{R}^2$?

I don't understand this sentence; The segment $(a,b)$ can be regarded as both a subset of $\mathbb{R}^2$ and an open subset of $\mathbb{R}^1$. If $(a,b)$ is a subset of $\mathbb{R}^2$, it is not ...
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86 views

N-points compatification

I know that Alexandroff compatification is unique, and if the Alexandroff compatification of two spaces are not homeomorphic, then the spaces can't be. Does uniqueness stand in n point ...
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127 views

How to prove that the space $ \omega_1\times R $ has countable extent?

How to proof that the space $ \omega_1\times R $ has countable extent? The topological space $\omega_1$ is the first uncountable ordinal with order topology. A space $X$ has countable extent if ...
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359 views

Locally Path Connected space that is not Path Connected.

Can you give me an example of a space that is locally path connected but not path connected, if it exists ?
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190 views

Generating the Sorgenfrey topology by mappings into $\{0,1\}$, and on continuous images of the Sorgenfrey line

Show that the topology of the Sorgenfrey line can be generated be a family of mappings into a two-point discrete space. Verify that the Sorgenfrey line can be mapped onto $D(\aleph_0)$ but cannot be ...
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165 views

does there exist a continuous function

$A_1=\{ \text {closed unit disk in plane}\}$ $A_2=\{(1,y):y\in \mathbb{R}\}$ $A_3=\{(0,2)\}$ We need to confirm: there exist always a continuous real valued function $f$ on $\mathbb{R}^2$ such that ...
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97 views

How to verify this function is continuous?

Recently, I'm reading a paper "Spaces with a regular Gδ-diagonal" of A.V.Arhangel’skii's. I can't understand the function $d$ in the example 9 is continuous. Could someone help me? Thanks ahead:)
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229 views

Definition of product of uniform spaces

In Wikipedia and PlanetMath product of uniform spaces is defined as the weakest uniformity on the Cartesian product making all the projection maps uniformly continuous. But Springer's encyclopedia ...
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371 views

The topology on $\mathbb{R}$ with sub-basis consisting of all half open intervals $[a,b)$.

Let $\tau$ be to topology on $\mathbb{R}$ with sub-basis consisting of all half open intervals $[a,b)$. How would you find the closure of $(0,1)$ in $\tau$? I'm trying to find the smallest ...
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125 views

The relative topology with respect to two spaces is the same.

In Bert Mendelson's "Introduction to Topology" p. 159, i read the statement "A topological space $C$ can be a subspace of two distinct topological spaces $X$ and $Y$. In this event the relative ...
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872 views

Prove that every convex region is simply connected

Prove that every convex region is simply connected Could anyone help?
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72 views

Topological Circle as a Quotient (trivial quesion)

Can someone explain me why does the unit circle can be expressed as the following quotient : $ \mathbb{R} / \mathbb{Z} $ ? Thanks !
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58 views

Topology carried over by a mapping of a topological space into a set.

Let $f:X \rightarrow Y$ be a mapping of a topological space $X$ into a set $Y$. Let $J$ be the topology on $X$. What exactly is "the topology carried over to $Y$ by $f$"?
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254 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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47 views

Show that $(X,T)$ is $T_2$

Can someone please help me with this problem? Assume that $(X, T)$ is a topological space and that $B$ is a basis for the topology $T$. Show that $(X, T)$ is $T_2$ if and only if for all $x$ and $y$ ...
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293 views

Basic proof that $T$ is a topology.

Question: Let $T = \{U \subseteq \mathbb{R} : U = \emptyset \text{ or } U = \mathbb{R}\text{ or } U = (−\infty, a) \text{ for some } a \in\mathbb{R}\}$. Prove that $T$ is a topology on ...
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229 views

If X is regular space and Y,a dense locally compact subspace, then Y is open?

prove or disprove: If X is regular space and Y,a dense locally compact subspace, then Y is open.
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411 views

What is the relation of basis in linear algebra and basis in topology?

In linear algebra and topology ,it all has the concept basis,but I can not construct the relation of them,could you explain the relation of two basis,such as the basis in linear algebra is special ...
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161 views

Intersection in compact topological spaces?

Set $X$ to be a compact topological space. Let ${V_\alpha}$ be a system of the closed subsets of $X$ where $\bigcap\limits_{\alpha\in ℤ}{V_\alpha}≠\emptyset$ ($\alpha$ is finite). Show ...
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143 views

Quotient by Homeomorphisms Produces an Open Map

If a group $G$ acts on a topological space $X$ by homeomorphisms, why is the quotient map $X \to X/G$ necessarily an open map?
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261 views

Does a subset consisting only of isolated points have a limit point?

In a topological space, if all elements of a subset are isolated, will the subset have any limit point? If it does have a limit point, then the limit point must of course not belong to the subset. ...
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169 views

When is $[0,1]^K$ submetrizable or even metrizable?

Let $I=[0,1]$ and $K$ is a compact space. Then could the function space $I^K$ be submetrizable, even metrizable? In other words, in general, if $I^A$ can be submetrizable (metrizable) for some space ...
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158 views

Order topology is the same as the metric topology on $\mathbf{R}$?

The wikipedia article for the real line says that the order topology and the metric topology of the reals are the same. What is the explanation that these two topologies are in fact identical?
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533 views

Is it true that under the Zariski topology, a subset is dense if and only if it is a nonempty open subset

Is it true that under the Zariski topology, a subset is dense if and only if it is a nonempty open subset? I know this is true in one direction, i.e., any nonempty open subset is dense, but how ...
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601 views

Confusion about the boundary of connected components

Let $C$ be a connected component of $X\subset\mathbb{R}^n$. I want to prove of disprove that $\partial{C}\subset\partial{X}$ (where $\partial{A}$ means the boundary set of $A$). In metric space, I ...
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89 views

an explicit example of a function with a local strictly maximum dense set

This problem looks very difficult )= Construct a continuous function, such that it set of strictly local maximum points, is the set of rationals.
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227 views

Show that rays of the form $(-\infty, a)$ and $(b, \infty)$ ; $a,b \in \mathbb R$, are a sub-basis for the standard topology on $\mathbb R$?

Show that rays of the form $(-\infty, a)$ and $(b, \infty)$ ; $a,b \in \mathbb R$, are a sub-basis for the topology generated by open intervals of $\mathbb R$ on $\mathbb R$? I'd just like to know if ...
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277 views

Extremally disconnected space

Can one view $\ell_{\infty}^{4}$, $\mathbb{R}^{4}$ equipped with the $\ell_{\infty}$-norm, as a space of continuous functions on some extremally disconnected space? and what would the extremally ...
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255 views

$\{ (x,y) \in \mathbb R^{2} | x>0, y \in \mathbb R \}$ not clopen?

Let $S_{1} = \{ (x,y) \in \mathbb R^{2} | y \geq \frac{1}{x}, x> 0 \}$ and $S_{2} = \{ (x,y) \in \mathbb R^{2} | x = 0, y \leq 0 \}$. Now $S_{1} + S_{2} = \{ (x,y) \in \mathbb R^{2} | x > 0, y ...
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116 views

Different topologies $\Rightarrow$ different neighborhood bases?

Can someone give me a proof or a counterexample to the following: Given two different topologies over the same set, the neighborhood bases of these two topologies have to differ in at least one point. ...
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Continuous function and open set

I'm trying to prove: If $f:A\subset \mathbb{R}^m \to \mathbb{R}^n$ and $A$ is open, then the following statements are equivalent: 1) $f$ is continuous on $A$. 2) $f^{-1}(V)$ is open in ...
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449 views

Hausdorff, locally connected and locally compact space exercise

Let $X$ be a Hausdorff, locally connected and locally compact space. Let $U$ be a connected subset of $X$ and let $x,y \in U$. Prove there exists a compact connected subset $T$ of $U$ such that $T$ ...
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202 views

lie groups and topology

Is there a relationship between Lie groups and topology and is there a succinct explanation that can be provided? Is there a good online reference that discusses this.
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711 views

Complement of all finite subsets of $\mathbb{R}$

I was skimming through the topology book recommended through this question and I came across a question that I apparently solved incorrectly. It's question 2.5 on page 12. Repeated here: Let $X$ ...
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369 views

Product of connected sets, proper subsets

Let $X,Y$ be connected sets and $A,B$ proper subsets of $X$ and $Y$ respectively. Then it can be shown that the space $ (X \times Y) \setminus (A \times B)$ is connected. This is an problem from ...
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319 views

Help with definition of n-dimensional smooth manifold

Again, I am reading this. I am finding it a bit difficult to understand the definition of n-dimensional smooth manifold. Now, $\{U_a; x^1_a, x^2_a, ..., x^n_a\}$ ...
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509 views

Prove this set is open

Let $A \subset X$ be closed and $U \subset A$ open in $A$. Let $V$ be any open set in $X$ with $U \subset V$. Prove $U \cup (V \setminus A)$ is open in $X$.
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213 views

When does a subbase of a base generate the same topology?

Suppose that $\mathcal{B}$ is a base for a topology on a space $X$. Is there a nice way of thinking about how we can modify $\mathcal{B}$ (for instance, to simplify computations) without changing the ...
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1answer
200 views

Sufficient condition for being measurable?

I'm playing around with a collection of subsets of $\mathbb{R}^n$ let's call them $X_i$. What I want to know is, is the following condition sufficient for some $X_i$ being measurable? Almost all ...
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150 views

Relationship between topology of a compact group and the topology of its profinite completion

Suppose $G$ is a compact topological group. We can construct the profinite completion of $G$; let's call this $\Gamma$. My questions are: 1) Assuming that we know nothing about the (original) ...
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72 views

Suppose every convergent Sequence has a unique limit point in space $X$, then $X$ is Hausdorff

My attempt Let $a, b \in X$ are two distinct points . we will show that there exist two open sets $G_1$ and $G_2$ such that $a \in G_1$ and $b \in G_2$ and $G_1 \cap G_2 = \phi$ or there exists $n ...
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53 views

Does a proper map have to be continuous?

In Pollack's differential topology, the proper map is defined by the preimage of every compact set is compact. Here it doesn't require the map to be continuous. However, in his following claim, to a ...
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52 views

Proof of Cantor's Intersection Theorem

I am going through metric spaces by Michael Searcoid. The text proves the Cantor's Intersection theorem as shown in the image below. I understand the proof. However, just one thing, I am a little in ...
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61 views

Prove that General Linear Group is a topological subgroup.

First of all for $\mathbb{R}$ in my book it is written that: "$GL(n,\mathbb{R})$ is an open subset of euclidian $n^2$-space and that is the topology is given. Matrix multiplication is given by ...
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Hausdorff space definition in terms of disconnected subsets

I've read the definition of a topological Hausdorff space (two distinct points have disjoints neighbourhoods) and of a disconnected space (it is the union of two disjoint nonempty open sets) Now I ...