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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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the discrete topology of an infinite set

Hi I am very new to topology and was wondering how to solve the following problem : Let X be an infinite set and τ a topology on X. If every infinite subset of X is in τ, prove that τ is the discrete topology. I have trouble understanding why this…
Jip Helsen
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Topology, Normal Spaces - Proving a function is continuous knowing that any composition is continuous.

my question is regarding topology. Given two topological spaces $X, Y$ and we know that $Y$ is $T_4$. Let $f:X \rightarrow Y$ be a function such that for any continuous function $\phi:Y \rightarrow \mathbb{R}$, the composition $\phi \circ f$ is…
Nadav Kalma
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Question about a base for a topology

Actually, this is only a clarification about the definition of a base for a topology. In the book of Dshalalow entitled "Real Analsysis: An Introduction to the Theory of Real Functions and Integration", CRC Press LLC, USA, 2001, at p.115, defined…
Juniven Acapulco
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Do homeomorphisms preserve topological structure?

I think something is missing in the definition of homeomorphism I saw. It just said it maps the collection of open sets to the collection of open sets in a bijective way. What exactly makes this preserve topology? I can think of weird situations…
Emil
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Is this a correct visualization of the topological quotient?

I was trying to come up with a visual intuition of the smash product of two topological spaces, and ended up understanding it as the result of the following process: 1. start with the topological product, 2. cut along the wedge product of the two…
Mehmet Ates
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Sphere-sphere intersection is not a surface

In my topology lecture, my lecturer said that when two spheres intersect each other, the intersecting region is not a surface. Well, my own understanding is that the intersecting region should look like two contact lens combine together,back to…
Idonknow
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Is there a topology which makes $[0,1)$ compact and Hausdorff?

Consider the interval $X= [0,1)$. Is there a topology which makes the interval Hausdorff and compact? My intuition tells me that such a topology cannot be found. I have attempted to prove that, if $\{a_n\}_n$ is a sequence which approaches $1$ from…
francescoriccardocrescenzi
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Is the extension of a continuous function on a subspace to the closure of its domain unique?

The following problem is from Munkres's Topology (exercise 13 in section 18 "Continuous Functions"; page 112, 2nd edition). Let $A \subset X$; let $f:A \to Y$ be continuous; let $Y$ be a Hausdorff. Show that if $f$ may be extended to a continuous…
Jack Wislly
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Is it possible to partition interval [0,1] into two closed sets with empty interior?

Consider the Eucidean topology over $\mathbb{R}$. Consider the unit interval [0,1]. I want to partition this interval into sets $A$ and $B$ such that $A \cup B =[0,1]$ Both $A$ and $B$ are closed and both $A$ and $B$ have an empty interior. It…
ak7019
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Every open subset of $\mathbb{R}$ can be expressed uniquely as a disjoint union of open intervals. Does this generalize to $\mathbb{R}^n$?

I know that every open subset of $\mathbb{R}$ can be expressed uniquely as a disjoint union of open intervals. Further, only countably many intervals feature in any such decomposition. Supposing we replace $\mathbb{R}$ with $\mathbb{R}^n$, and 'open…
goblin GONE
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Unraveling the various definitions of $k$-space or compactly generated space

There are multiple (incompatible) definitions of "compactly generated" or "$k$-space" in the literature. For a sample, see the various references mentioned in this question. See also the discussion here and here for example. One starts with a…
PatrickR
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Does $X\times \mathbb{R}$ characterize $X$ as topological space?

Let $X,Y$ be topological spaces s.t. $X\times\mathbb{R}\simeq Y\times\mathbb{R}$. I want to know if $X\simeq Y$?
Yos
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How to prove two topologies $\mathcal{T}_1,\mathcal{T}_2$ are not equal

Let $C[0; 1]$ be the set of all continuous real-valued functions on $[0; 1]$. (i) Show that the collection $M$, where $M = \{M(f,\varepsilon ) : \text{$f\in C\left[0; 1\right ]$ and $\varepsilon $ is a positive real number}\}$ and $M(f,\varepsilon)…
Jebei
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is the existence of a coarsest topology nontrivial

I understand what the definition of "the coarsest topology satisfying some property x" is, but my question is, shouldn't the existence of such a topology be nontrivial? For example, let $X = \{ a, b, c \}$, and let our property be "has four…
Abced Decba
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Topology of rationals as subspace of the reals vs rationals with euclidean topology

I am trying to compare different topologies on the set of rationals. I feel like the following two are distinct (i.e. some sets are open in one and not in the other), but I can't write a formal enough argument unfortunately. Any feedback is very…
Alf
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