# Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

6,632 questions
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### Existence of a Zero of a Continuous Map

Suppose that $f: \mathbb{R}^n \to \mathbb{R}^n$ is a continuous map, satisfying $$\langle x,f(x) \rangle \geq 0, \forall x\in S^{n-1}=\{x\in \mathbb{R}^n: \Vert x\Vert =1 \} .$$ Prove that there ...
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### How to prove or disprove the following set is compact?

$P$ is set of all real polynomial in one variable. Define $$d(p,q)=\max\{|p(x)-q(x)|:x \in[0,1]\}$$ and $$K=\{p \in P,d(p,0)\leq 1 \}$$Prove or disprove $K$ is compact. I think the only way to prove ...
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### Proving that $\{ x \in \mathbb{R}^n : |x| = 1, x\geq 0\}$ is homeomorphic to $\overline{B_1(0)} \subset \mathbb{R}^{n-1}$

I want to show that $A= \{ x \in \mathbb{R}^n : |x| = 1, x\geq 0\}$ is homeomorphic to $\overline{B_1(0)} \subset \mathbb{R}^{n-1}$, where $\overline{B_1(0)}$ is the closed unit ball. I was thinking ...
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### Cluster Point vs Limit Point

Are there any differences between a cluster point and a limit point? If yes, is every cluster point is also a limit point?
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### Is the following subset of $\mathbb{R}^2$ open, closed or neither?

I am given the subset $A=\{(x, \sin(1/x)) | x>0\} \bigcup \{(0,y) | y \in [-1,1] \}\subset \mathbb{R}^2$ equipped with the standard Euclidean metric I came to the conclusion that it is not open ...
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### $X$ is c-closed iff every countably compact subset of $X$ is closed

I need help to understand definition of c-closed. In here https://www.sciencedirect.com/science/article/pii/0166864180900279, author said that $X$ is c-closed iff every countably compact subset of $X$ ...
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### Topology of the torus

Theorem: There is no open covering $T^k=U_1\cup...\cup U_k$ of the $k$-torus such that the map $$H_1(V,\mathbb{Z}) \rightarrow H_1(T^k,\mathbb{Z})$$ has rank at most $(i-1)$ for every component $V$ ...
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### Prove continuity from preservation of sequential limits but do not use $f(\bar{A}) \subseteq \overline{f(A)}$

Munkres Topology: Theorem 21.3. Let $f : X \rightarrow Y$. If the function $f$ is continuous, then for every convergent sequence $x_n \rightarrow x$ in $X$, the sequence $f(x_n)$ converges to $f(x)$...
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### Given a closed disc staying in an open set in $\mathbb{C}$, is there a larger open disc within the open set which contains this closed disc?

Let $U$ be open and $D \subset U$ be a closed disc. In the book Complex Analysis by Stein (P.39 Corollary 2.3), he claimed in the proof that there exists a slighly larger open disc $D' \subset U$ such ...
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### Definitions of the 'limit points'

I knew that the definition of limit points $x$ of a subset $E$ of $\mathbf{R}^d$ is : For any open ball $B$ containing $x$, $(B\setminus \{x\})\cap E\ne\emptyset$ Or something like these arguments ...
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### Convex Homotopy

Suppose $f , g : X \to U \subset \mathbb R^2$ are two mappings from a topological space $X$ to a convex set $U$. Prove that $f$ and $g$ are homotopic, using only the definition of the product mapping....
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### Conditions for continuity of composition

Let $f:X\rightarrow Y$, $g:Z \rightarrow Y$, and $h:Z \rightarrow X$ be functions such that $f \circ h = g$, where $X, Y$ and $Z$ are topological spaces. What are some weak conditions that we must ...
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### If there exists an open set $U$ in $X$ such that $A = Y \bigcap U$ then $A$ is open in $Y$

Let $Y$ be subspace of a metric space $X$. Show that $A \subset Y$ is open in $Y$ if and only if there exists an open set $U$ in $X$ such that $A = Y \bigcap U$. My Try: Let $A$ be open in $Y$. Then ...
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### Showing set $A = [0,1)$ in a discrete metric is open and closed

Let $X = \mathbb R$ with metric $\rho_0 : X \times X \to \mathbb R$ defined by $$\rho_0 (x,y) := \begin{cases} 1, & x \ne y;\\ 0, & x = y; \end{cases}$$ - $\bullet$ If $A = [0,1)$, then ...
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### Let $F$ be a closed subset of $(X,d)$ and suppose that $\inf\{d(x,z): z \in F\} = 0$ for some $x \in X$. Prove that $x \in F$.

All the limit points of $F$ are contained in $F$ since $F$ is closed. $d(x,z)$ is a set of real numbers. It is obvious that $\inf\{d(x,z): z \in F\} = 0$ for some $x \in X$. I have no idea how to ...
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### Boundary Sets are Closed

The following question is from Fred H. Croom's book "Principles of Topology" Prove that the boundary of a subset $A$ of a metric space $X$ is always a closed set. My attempt is as follows: ...
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### Showing set is closed

Consider the set {$\frac{1}{2},\frac{1}{3},...,0$}. Show that this set is closed. How would one approach such a question without pursuing an epsilon-delta kind of proof?
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### Proof check: proving a neighborhood is an open set?

I want to prove that a neighborhood is an open set by picking an arbitrary point in it and showing it's an interior point. On my final exam I couldn't think of a way to use the triangle inequality(...
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### Wedge Sum Embedding with Inclusions

Let $X$ and $Y$ be two disjoint topological spaces, $x_0\in X$, $y_0\in Y$ and we consider the Wedge Sum (the quotient of the union by the relation $x_0\sim y_0$). I want to proof that $\pi \circ i_X$...
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### Conditions on $A,B$ that are inherited by $A+B$.

Let $A,B$ be subsets of $\Bbb{R}$. Which of the following is false: If $A,B$ are bounded, then so is $A+B$. If $A,B$ are open, then so is $A+B$. If $A,B$ are closed, then so is $A+B$. If ...
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### Definition of embedded in topology

One of the questions in my topology homework starts with: Suppose $G$ is a graph and $L \subset G$ is an embedded circle. I have looked around and found lots of definitions for an 'embedding' but ...
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### The intersection of a descending sequence of open sets with a point in common has a nonempty interior.

If in a topological space $X$, we have a descending infinte sequence of open sets, $\{V_n\}_\mathbb N$ ($V_{n+1}\subseteq V_n$) such that for a fixed $z\in X$, $z\in V_n$ for each $n\in \mathbb N$. ...
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### Prove that $\Delta = {\{(x,x);x}$ in ${X\}}$ in $X\times X$ can be written as an intersection of open sets iff X is $T_1$.
Prove that $\Delta = {\{(x,x);x}$ in ${X\}}$ in $X\times X$ can be written as an intersection of open sets iff X is $T_1$. The topology in $X\times X$ is the product topology. $T_1$ is the first ...
0 down vote favorite Let $(𝑋,𝒯)$ and $(𝑋',𝒯\ ')$ be topological spaces. Suppose $𝑓: 𝑋 → 𝑋'$ is an onto open map that has the extra property that $𝑈′$ open in $𝑋'$ implies that inverse of \$𝑓(...