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.
59,721 questions
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What is the topology on the set of all plays of a game?
I read the wikipedia page on the "Angel Problem"(https://en.wikipedia.org/wiki/Angel_problem). It stated that there must be a winning strategy for either the angel or the devil because the &...
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Existence of continuous curve contained in an level curve.
Let $f:[0,1]\times \mathbb{C}^{m \times n}\to \mathbb{C}^{m \times n} $ be a continuous function such that for each $t\in [0,1]$, there exists a $Z_t$ such that $f(t,Z_t)=0$. Can we find a continuous ...
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A product of a quotient map with an identity
Let $Y = \mathbb{Q}/\mathbb{Z}$ be the quotient space of the rationals obtained by collapsing the subset $\mathbb{Z}$ to a point, and let $q \colon \mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$ be the ...
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Topology (Munkres) Let $p : E \to B$ be a covering map, where $E$ is path-connected and $B$ is simply connected. Then $p$ is a homeomorphism. [closed]
Let
$p : E \to B$
be a covering map, where $E$ is path-connected and $B$ is simply connected.
Then $p$ is a homeomorphism.
Can someone please check
Since $p$ is a covering map, it is surjective, ...
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In Topology, is there a Name for the Closure of the Complement?
Let $S$ be a subset of a topological space $X$. The interior of $S^c$ is called the exterior of $S$. Is there a name for the closure of $S^c$? If so, in what sources has that terminology been used?
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Convergence in Abstract Topological Spaces (Munkres) [closed]
Munkres in the discussion before Hausdorff spaces defines convergence in abstract topological spaces like ‘a sequence of points $x_1, x_2,…$ converges to $x$ provided that, corresponding to each nbd $...
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Invariant elements of dense sub-module is dense
Let $A$ be complete topological abelian group (in my consideration, $A$ is actually an $\mathbb{F}_p$-algebra). Let $M$ be a dense subgroup of $A$. Suppose $\mathbb{Z}$ acts on $A$ and $M$ ...
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one detail of homeomorphism
I have some problem reading this answer:
Now let $q: \mathbb{R}_{\geq 0} \rightarrow S^1$ be the restriction
of $p$ to the non-negative reals and consider the pre-image
$q^{-1}(U)$ of our small open ...
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This proof of Tychonoff's Theorem seems to assume contradictory statements
The proof of Tychonoff's Theorem from Gamelin and Greene's "Introduction to Topology" states:
Let $X=\prod X_\alpha$, where each $X_\alpha$ is compact. Let $\mathcal{D}$ be a family of ...
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The "naive" vs. "true" homotopy category
It has been a while since I took algebraic topology or read about homotopy theory, so I wanted to refresh my understanding of the definition of the homotopy category (particularly the pointed version)....
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Prove the derived set $L$ of a subset $A$ of a metric space is closed using the sequential definition.
Prove the derived set $L$ of a subset $A$ of a metric space is closed using the sequential definition.
To do so if I proved if a point is the limit point of the derived set, it's a limit point of ...
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How can we prove that the composition of two isometries is bijective? [closed]
Let $A$ and $B$ be two isometries in $\mathbb{R}^n$. Prove that the composition $ A \circ B$ is bijective.
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Compute Fundamental gourp using Vam Kampen Theorem [duplicate]
Compute the fundamental group of the complement of the three coordinate axes in $\mathbb{R}^3$, giving explicit generator(s).
Let $X$ denote the space in the problem, $S=\mathbb{S}^2-\{\text{6 points ...
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When is a metrisable Souslin space a Polish space?
Recall that a Polish space is a separable topological space which can be metrised by a complete metric, and that a Souslin space is any Hausdorff topological space $S$ admitting a continuous ...
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Generalized Heine-Borel Theorem - Understanding proof
From T. B. Singh's Introduction to Topology book:
Theorem 5.1.16 (Generalized Heine-Borel Theorem) A closed and bounded subset $A$ of the Euclidean space $\mathbb R^n$ is compact, and conversely.
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