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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A basic question on inverse image of an open set being open

Inverse image of every open set (in "range" or "co-domain" which one is true ??) of a continuous function must be open. if it is co-domain then how openness of a set which is not part of a mapping ...
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2answers
39 views

Is $(\mathbb R,\tau)$ compact?

Let us consider $\tau=\{G\subset \mathbb R: \mathbb R\setminus G$ is compact set in ($\mathbb R,\tau_u)\}$, where $\tau_u$ denotes the usual topology on $\mathbb R$. Then $\tau$ is a topology coarser ...
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65 views

Topology question??

Suppose we have a surface $S$. Also, suppose we remove $2$ discs from the surface $S$ and we glue the boundary circles of these two discs together. Is the result a surface?? My believe is that it is ...
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51 views

On Homeomorphim

Let $X$ and $Y$ be topological spaces. I want to show that if $f:X\to Y$ is continuous then $\phi:X\to G(f)$ defined by $\phi(x)=(x,f(x))$ is open and continuous, where $G(f)$ is the graph of the ...
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31 views

Confusion about this assertion

I have seen this result true in general ? Every zero set is $G_\delta$-sets. As I know in Normal spaces, every closed $G_\delta$ set is zero set. Thx in advance
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57 views

Prove that $\exists a \in X$ s.t. $\bigcap ^{\infty} _{n=1} F_n = \{a\}$

Let $F_1, F_2,...$ be nonempty closed subsets of a complete metric space $X$ and suppose $F_1 \supseteq F_2 \supseteq ...$ and $\def\diam{\operatorname{diam}}\lim_{n\to \infty} \diam(F_n) = 0.$ Prove ...
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96 views

Questions about the Stone-Čech compactification

A compactification of $X$ is a pair $( h, Y)$ where $Y$ is a compact space and $h\colon X ‎\to Y$ is an embedding such that $h(X)$ is dense in $Y$. According to definition of mentioned ...
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63 views

Showing a set is open in $\mathbb{R}^2$

Show that $\mathbb{R}^2 \setminus \{(0,0)\}$ is open in $\mathbb{R}^2$. I'm not sure if this is obvious I can't give enough details. Every point in $U=D(v,||v||)$ has a $\epsilon$-neighbourhood ...
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47 views

Order Topology on $\mathbb{Z_+}$

Why is the order topology on $\mathbb{Z_+}$ a discrete one? I understand that the discrete topology will have all subsets of $\mathbb{Z_+}$ which means that all subsets of $\mathbb{Z_+}$ are open ...
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38 views

Whether $U\setminus C$ is open if $C$ is closed and $U$ is open

If $C$ is a closed set and $U$ is an open set, then is $U\setminus C$ open in an arbitrary metric space? I don't think this holds in the discrete metric space.
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50 views

Connectedness being a topological property?

So in Pugh's "Real Mathematical Analysis", he asserts that: If $M$ is connected and $M$ is homeomorphic to $N$, then $N$ is connected. If I wanted to show that the intervals $(a, b)$ and $(a, ...
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151 views

If S is a closed set, prove that $\partial S$=$\partial(\partial S)$

If S is a closed set, prove that $\partial S$=$\partial(\partial S)$. I'm trying to prove this using the equation $\partial S$=cl($\partial S$)=int($\partial S$)$\cup \partial(\partial S)$, then we ...
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70 views

A set is called sequential - closed if it contains all its sequential limit point.

A set is called sequential-closed if it contains all its sequential limit point. A set is called sequential-open if it is a sequential neighborhood ($N$ is a sequential neighborhood if whenever $x_n$ ...
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55 views

Closure of a Set

If $A=[1,2]$, then it's closure is $[1,2]$. Why is this true? My understanding is that closure A is the set containing element $x\in A$ such that for all open neighborhood $U$ of $x$, $U\cap A \neq ...
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274 views

Show a set is open

Show that the set A = {(x,y): y < 0} is open. I understand that to do this, i need to take an open ball centered at an arbitrary point in A with a positive radius and show it is contained in the ...
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72 views

Is the set of connected components of a basis another basis in $(\mathbb{R}^2,\tau_u)$?

I'm trying to prove the following result of basic topology: Let $\mathbb{B}$ be a basis of the standard topology in $\mathbb{R}^2$: $(\mathbb{R}^2,\tau_u)$. Let $\mathbb{B}'$ be the set of the ...
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27 views

Try to prove a claim

Here is a claim: Let $X$ be a space with $d(X)=\mathfrak c$. If $X$ is compact and homogeneous, then for any open set $U$ of $X$ $d(U)=\mathfrak c$. A topological space is homogeneous if for any ...
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50 views

Is the following an example of one point compactification?

How do I prove that the space $X = \left\{\frac 1n : n \in \mathbb Z\right\}$ is not compact but $X \cup \{0\}$ is compact ?
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471 views

Open Cover Having No Finite Subcover

Information: Consider the open cover $\mathscr{F}=\begin{Bmatrix}\pmatrix{0,5-\frac{1}{n}}:n\in\mathbb{N}\end{Bmatrix}$ for the set $(0,5)$. Question: Is ...
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45 views

A question about function on product spaces

Let $\{X_i:i\in I\}$ be a family of topological spaces. Is the function $f$ from $X=\prod\{X_i:i\in I\}$ to $X_K=\prod\{X_k:k\in K\}$, Where $K$ is a finite subset of $I$, continuous and open? The ...
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575 views

The product of Hausdorff spaces is Hausdorff

I'm confused how it can be true that the product of an infinite number of Hausdorff spaces $X_\alpha$ can be Hausdorff. If $\prod_{\alpha \in J} X_\alpha$ is a product space with product topology, ...
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95 views

Finding closure of a set

Let us consider $\mathbb{R}$ with the usual topology on it and let $x\in \mathbb{R}$. What will be the closure of $\lbrace x+r: r\in \mathbb{Q} \rbrace$?
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27 views

Use the fact that $P \# K \cong3 P$ to show that $kP \#nT \cong mP$ for some $m$, as long as $k>0$.

Use the fact that $P \# K \cong3 P$ to show that $kP \#nT \cong mP$ for some $m$, as long as $k>0$. Express $m$ as a function of $k$ and $n$. Here $P$ denotes projective plane, $T$ denotes torus. ...
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211 views

Continuous one-to-one mapping of a compact space

Prove that every continuous one-to-one mapping of a compact space is topological. Does this problem statement refer to a mapping of a compact space to itself? If so, suppose the mapping is ...
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3answers
164 views

Topological mapping of open disk onto whole plane

Construct a topological mapping of the open disk $|z|<1$ onto the whole plane. I represent $z=re^{i\theta}$. I thought about the bijection from $(0,1)$ to $(0,\infty)$, which is given by ...
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1answer
37 views

Deformation retract needs to be smooth?

So I am not quite sure that why none of these three is a deformation retract - is that because of the corners? But I don't remember deformation retract rely on smooth criteria, instead, on continuous ...
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1answer
54 views

Product topology

True or False: There can not exist topologies $\tau, \tau^{'}$ on an infinite set $X$ such that the product topology for $(X,\tau)$ and $(X,\tau^{'})$ coincides with the cofinite topology on $X\times ...
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1answer
97 views

A set $A$ is closed iff $\operatorname{fr}A\subseteq A$.

A set $A$ is closed iff $\operatorname{fr}A \subseteq A$. MY attempt: suppose $A$ is closed and let $x \in\operatorname{fr}A$. We must show $x \in A$. If $x \notin A$, then $x \in \mathbb{R}^n ...
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2answers
89 views

Homeomorphism between 2 Sets

How can I show that $SL(n,R)$ is homeomorphic to $SO(n,r)$x$R^(n^2+n-2)/2$]? Here $SO(n,R)$ denotes the set of all real orthogonal nxn matrices with positive determinant. Thanks for any help.
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76 views

Proof on if a set is discrete

I would like to know how well I answered the following proof: was it concise? Was it elaborate/rigorous? Did I use incorrect notation? I would also like to know if the set is a $T_{1}$ space, such ...
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1answer
93 views

Elementary proof that $\mathbb{R}^n$ minus hyperplane is not connected

I was wondering if an elementary proof is possible of the following fact, i.e. without using Invariance of Domain, Jordan Curve Theorem, etc. Prove that if $H$ be a hyperplane in $\Bbb R^n,$ then ...
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3answers
104 views

compactness of topological space

i would like to understand easily notation of compact space,i had read that space is compact if it is closed and bounded,fr example following link says that The closed unit interval $[0,1]$ is ...
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1answer
45 views

definition of accumulation point

is correct the following definition? -- "let $ s \in \mathbb{R} $ and $ T \subseteq \mathbb{R} $, $ s $ is accumulation point for $ T $ if $ \forall S \in \mathcal{U}(s)((S-\{s\})\cap T \neq ...
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1answer
40 views

Is the image of a continuous idempotent necessarily homotopic to the original space?

Let $f$ be a continuous self-map of a topological space $X$ such that $f\circ f=f$. Is it true that $X$ is homotopic to its image $f(X)$?
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1answer
64 views

locally compact and compatification

As we know a topological space $X$ is said to be locally compact at a point $ x \in X $ if $x$ has a compact neighbourhood in $X$. $X$ is called locally compact if it is locally compact at every ...
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68 views

infinite regular cardinal

Let $(X,\tau)$ be a KC non-compact space. Then there is a discrete subset $D \subseteq X$ such that $\overline D$ is not compact. Furthermore there is an ultrafilter $F$ in $X$ such that $ D \in F $ ...
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77 views

Question about the definition of subbasis.

Let $\mathcal{S}_{1}$ be the collection of all circles in the plane which have their centres on the $X$-axis. If $\mathcal{S}_{1}$ is a subbasis for a topology $\mathcal{T}_{1}$ on $\mathbb{R}^{2}$, ...
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1answer
57 views

Every KC space is not Katetov - KC.

A $KC$-space $(X,\tau)$ is said to be Katětov-$KC$ if there is a minimal $KC$-topology $\sigma\subseteq\tau$. A space is $KC$ if each compact space is closed. I know that not every $KC$ space is ...
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1answer
42 views

Example of sequential space which is not Fréchet

Definition: $X$ is a sequential space if, whenever $A\subset X$ and $A$ is not closed, there is a sequence $\{a_n:n∈ω\}⊂A$ such that $a_n→y$ for some $y\in A^c$. Is there any example to show ...
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1answer
85 views

KC-spaces and US-spaces.

A topological space is called a US-space provided that each convergent sequence has a unique limit. A topological space is called a KC-space provided that every compact subset is closed. So ...
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3answers
33 views

If $f_i(X)$ is connected for all $i=1,2,…,n$ then $X$ is connected.

For each $i\in\{1,...,n\}$, consider the map $f_i:\mathbb{R}^n\to \mathbb{R}$ defined by $f_i(x)=x_i$ for all $x=(x_1,...,x_n)\in\mathbb{R}^n$. I would like to know if the following statement is true ...
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189 views

Show that $f$ is discontinuous.

Let the sequence of function $f_{n}=\sqrt[2n+1]{x}$ (for $x\geq 0$). I've shown that it converges pointwise to $f$, that is $$\lim_{n\to\infty}f_{n}(x)=f(x)=\left\{\begin{matrix} 0 ...
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1answer
124 views

Intro to Topology Mendelson

I'm self studying intro to topology by Mendelson and I'm stuck on a book problem. The problem is, Let $Y$ be a subspace of $X$ and let $A\subset Y$. Denote the $\operatorname{Int}_X(A)$ as the ...
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170 views

Clustering of Cofinally Cauchy nets

If $(X,d)$ is a metric space in which every Cofinally Cauchy sequence clusters. Does this imply every Cofinally Cauchy net clusters in the space?
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189 views

An example of disconnected set

I have just learned the definition of connectedness and wikipedia says (here, https://en.wikipedia.org/wiki/Arc_connected#Arc_connectedness) $(0,1)\cup \left\{ 3 \right\}$ is disconnected. I need a ...
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133 views

Uniformly equivalent metrics and the metric on a countable product space

Two metrics $d_1, d_2$ on a set $X$ are called uniformly equivalent, iff for every $\varepsilon > 0$ there exists $\delta_1, \delta_2$ such that $$ d_1(x,y) < \delta_1 \Rightarrow d_2(x,y) < ...
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1answer
42 views

Step in this proof unclear

I have a question about the following proof that is given here: Proofwiki Why does the intersection of all $B_x$ only contain $\{x\}$? So how do I see that there is nothing else in this intersection? ...
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102 views

Any homeomorphism from the boundary of a disc to itself can be extended to a homeomorphism of the whole disc

Let $A$ be a disc. We are given a homeomorphism $f:∂A\to ∂A$ and we want to extend it to a homeomorphism $g:A\to A$. If it is true that the union of two homeomorphisms is a homeomorphism (which I ...
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145 views

Projection of closed set [duplicate]

Set $A \subset R^2$, set B is projection of A on x-axis. Do you know a counterexample to the statement: if A is closed, then B is closed.
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146 views

Can continuous real-valued function on a countably compact space attain its maximum and minimum?

It is said in my textbook that a continuous real-valued function on a countably compact space can attain its maximum and minimum. However, the proof is not given. I cannot make a proof, and have done ...