# 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.

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### $A \subset \mathbb{R}^n$ is bounded if and only if its closure $\overline{A}$ in $\mathbb{R}^n$ is compact.

I would like to prove that $A \subset \mathbb{R}^n$ is bounded if and only if its closure $\overline{A}$ in $\mathbb{R}^n$ is compact. This is problem 23.2 in Tu's book on manifolds. Here is my proof: ...
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### Prove that (R, \tau cof ) is topological space

Advanced topology Prove that (R, tau cof ) is topological space?
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### Relationships between completeness and closedness

Specifically. Suppose X is a metric space, A $\subseteq$ X .I want to show A is closed in X.
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### Is $\{(z_1,z_2) \in \mathbb{C}^2 :|z_1|=|z_2| = 1, z_1^m = z_2^n\}$ a circle?

Fix $m,n \in \mathbb{Z}\setminus \{0\}$ and consider the subspace $$A = \{(z_1,z_2) \in \mathbb{C}^2 :|z_1|=|z_2| = 1, z_1^m = z_2^n\}$$ of the $2$-torus. Is this space homeomorphic to a circle? ...
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### Relative topology on [0, 1]. Filters. Convergence.

Let $\mathscr F$ be the collection of subsets of $[0,1]$ which contain an interval $[\frac{1}{3},\frac{2}{3}]$. Use the relative topology on $[0,1]$ considered as a subspace of $\mathbb R$ with the ...
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### Torus inside $S^2\times S^2$

Suppose we have a torus $\mathbb{T}^2$ inside $S^2\times S^2$, i.e., there exists a continuous injective function $\phi:S^1\times S^1\rightarrow S^2\times S^2$. Is it always true that this torus can ...
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### a bounded sequence that converge weakly [closed]

let E be a normed vector space such that E is isomorphe to E**, let $(x_n)_n$ a bounded sequence on E ,prove that we can construct a subsequence of $x_n$ that converge weakly.
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### Open sets. Hausdorff Space.

Consider the topology on $[-1, 1] \times [-1, 1] \subset \mathbb R^2$ where the set is considered open if and only if it is empty, or contains $(0,0)$. Prove: This definition of open sets defines a ...
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### the additive of continuous maps [closed]

Let $X$ be a topological space. $f$ and $g$ are two continuous maps from $X$ to real number space $\mathbb R$. Prove that $f+g$ and $fg$ are also continuous maps from $X$ to $\mathbb R$.
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### Some examples to demonstrate compact support

I am an electrical engineer trying to learn some real analysis from the internet. Please provide an instructive example of a function with compact support and another function without compact support. ...
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### $\mathrm{Top}$ as a double category

In notes of Susan Niefields about double categories (http://www.tac.mta.ca/tac/volumes/26/26/26-26.pdf) one of the first examples of a double category is a double category structure on $\mathrm{Top}$, ...
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### Understanding the meaning of a topology being weaker than another.

In topology, a topology $T_1$ is said to be weaker than another $T_2$ if $T_1$ is contained in $T_2$. The reason for "weaker" is that there could be more convergent sequences in $T_1$ than ...
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### Proving $\partial (A \cap B) \subset \partial(A) \cup \partial(B)$ using cases

Let $A$ and $B$ be two subsets of a topological space $S$. I would like to prove that $$\partial (A \cap B) \subset \partial(A) \cup \partial(B)$$ where $\partial$ denotes the boundary of that set. I ...
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### Prove existence of disjoint open sets containing disjoint closed sets

Let $(X, d)$ be a metric space, and let $F_1$ and $F_2$ be disjoint closed subsets of X. Prove that there exist open subsets $U_1, U_2 ⊂ X$ so that $F1 ⊂ U1, F2 ⊂ U2$ and $U1 ∩ U2 = ∅$. Prove ...
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### Anti-isometry that permits existence of metric space that inverts relations of another metric space

Suppose we have two spaces with respective metrics $d_1$ and $d_2$ and an (anti-)isometry $f$ that maps objects from the first space to the second. I'm interested in the setting where the following ...
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### Relating Line and Circle Topologies

I'm trying to develop a better conceptual understanding for how a topology endows shape to a space. Contrasting discrete (too many open sets so points too far from each other), Hausdorff (just enough ...
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### Is there a name in English for this kind of compact set?

In French, they call “compact à bord régulier” a compact set $K \subset \mathbb{C}$ such that $K=cl(int(K))$ and $\partial K$ is a finite union of piecewise $C^2$, disjoint and orientable (relatively ...
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