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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.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

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On step-4 of Munkres' proof of the Urysohn lemma

In this question my reference is the book Topology by Munkres. The proof of Urysohn's lemma can be found in section 33. In the last step of the proof, the author says: Now we prove continuity of $f$. Given a point $x_{0}$ of $X$ and an open…
Sardines
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Why $\Bbb{R}$ and $\Bbb{R}^2$ are not homeomorphic.

Can I argue that if there exists a homeomorphism $$f: \Bbb{R} \rightarrow \Bbb{R}^2$$ Then subtracting a point should preserve connectedness by continuity of $f$, but then $\Bbb{R}$ minus the origin is disconnected while $\Bbb{R}^2$ minus the origin…
homosapien
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the quotient of a sphere by a contractible closed subset in it

let $A \subsetneq S^n$ be a contractible closed subset of the $n$-dimensional sphere. my visual intuition for the two-dimensional sphere tells me there should be a homeomorphism $S^n / A \cong S^n$. is it true? how can one prove this?
windfish
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Continuous open maps on compact sets are surjections.

Could someone help me to show that if $X\subset \mathbb{R}^m$ is compact, then every continuous open map $f:X\to S^n$ is surjective? This question was taken of an Analysis book (the subject of section is connectedness) Thanks.
Pedro
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Are there any non-compact topologies such that every real-valued continuous function has a maximum value?

I asked my topology teacher this and she didn't know. Does there exist a non-compact topology $E$ such that for every continuous function $f : E \to \mathbb{R}$, $\text{max}(f(E))$ exists?
Perry Ainsworth
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There is no injective continuous map from $(\mathbb{R}-\mathbb{Q})\times\mathbb{R}$ to $\mathbb{R}$.

I have to show that there is no injective continuous map from $(\mathbb{R}-\mathbb{Q})\times\mathbb{R}$ to $\mathbb{R}$. Let $Y=(\mathbb{R}-\mathbb{Q})\times\mathbb{R}$. I thought about doing something with connectedness and the image of a connected…
Fabrizio Gambelín
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Baire category and relative topology

The present problem is realted to the following posting in MSE. Recall that for a given a topological space $(X,\tau)$, a set $E\subset X$ is nowhere dense in $X$ if $\operatorname{Int}(\overline{E})=\emptyset$. A subset of $X$ that can be written…
Mittens
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What is a path in a topological space?

A path in a topological space X is a continuous function from the closed unit interval [0, 1] into X. What happens when the topological space is something more simple, for example given $X = \{ 1, 2, 3, 4\},$ consider the topology $\tau = \{…
apg
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Invariants of bitopological spaces

A bitopological space is a set $X$ with two fixed Hausdorff topologies $\tau_1, \tau_2$. In my case I am interested in the case where $\tau_1 \subseteq \tau_2$. Say that a bitopological space $(X, \tau_1, \tau_2)$ is compactly almost metric,…
Tomasz Kania
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An exercise about countable basis

In the book Munkres: Topology §30 I met the following problem: Show that if $X$ has a countable basis $\{B_n\}_{n \in \mathbb{Z}_+}$, then every basis $\mathscr{C}$ for $X$ contains a countable basis for $X$. [Hint: For every pair of indeces $n$,…
mcihak
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Is a function between topological spaces continuous if continuous on subspaces?

Assume that there are two topological spaces $X,Y$ and a function $f:X\rightarrow Y$. Furthermore, assume that there exists a collection of sets $\mathcal{B}$ such that $\bigcup_{B\in \mathcal{B}} B = X,$ and for each $B \in \mathcal{B}$ the…
JMill.
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Is it always possible to extend continuous functions defined on a *closed* subset of a locally compact Hausdorff space?

In the following lemma the authors used Tietze's extension to get $f_1$ and $g_1$. I know this version of Tietze, but it requires the subset to be compact not merely closed, i.e., continuous functions defined on compact subsets of a LCH space can…
sigma
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Non-trivial Topology

I can't understand the differences between a non-trivial topology and a trivial one. Whuat's the meaning of "non-trivial" topology? Is there a link with connection's properties? For example, could we say that a moebius strip has a "non-trivial"…
xino
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Regular open sets and semi-regularization.

In a Hausdorff space $(X,\tau)$, we can generate a coarser topology, say $\tau'$, by taking its base to be the family of regular open sets in $(X,\tau)$. (Semi-regularization of $(X,\tau)$) Given that it's already proven, how to we proceed to prove…
Nino
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Exercise 2.2.12 in Topology and Groupoids

The following is Exercise 2.2.12 is Ronald Brown’s Topology and Groupoids, which tries to characterize topological spaces using the relation “$A$ is contained in the interior of $B$”. Let $X$ be a non-empty set and $⊲$ a relation on subsets of $X$…
Jendrik Stelzner
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