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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.

57719 questions
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Find two closed subsets or real numbers such that $d(A,B)=0$ but $A\cap B=\varnothing$

Here is my problem: Find two closed subsets or real numbers such that $d(A,B)=0$ but $A\cap B=\varnothing$. I tried to use the definition of being close for subsets like intervals but I couldn't find any closed sets. Any hint? Thank you.
Basil R
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Is a torus a subset of $\mathbb{R}^3$ or $\mathbb{R}^4$?

Defining a torus $T$ as $S^1 \times S^1$, it should follow that $T \subseteq \mathbb{R}^4$. But you can also think of a torus as a bagel, which means it's a subset of $\mathbb{R}^3$. Can anyone clarify this point?
user78720
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4 answers

The definition of continuous function in topology

I am self-studying general topology, and I am curious about the definition of the continuous function. I know that the definition derives from calculus, but why do we define it like that?I mean what kind of property we want to preserve through…
89085731
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15
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3 answers

Homeomorphism of the real line-Topology

I need to show that any open interval is homeomorphic to the real line. I know that $f(x)=a+e^x$ will work for the mapping $f:R \to (a,\infty)$ and $f(x)=b-e^{-x}$ will work for the mapping $f:R \to (-\infty,b).$ Without using two functions, how can…
ccc
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Base of a topology

I am confused with the concept of topology base. Which are the properties a base has to have? Having the next two examples for $X=\{a,b,c\}$: 1) $(X,\mathcal{T})$ is a topological space where $\mathcal{T}=\{\emptyset,X,\{a\},\{b\},\{a,b\}\}$. Which…
Haritz
  • 563
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3 answers

Closure of continuous image of closure

Let $f:X \to Y$ be a continuous map between topological spaces and $A \subset X$. Is it true that $\overline {f( \overline A)}= \overline {f(A)}$?
Bernard
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"Every open cover admits an open locally finite refinement" - can this refinement always be realized in terms of basis sets?

Let $X$ be a paracompact topological space, let $C$ be an open cover, and let $\mathscr B$ be a basis for the topology. Does there always exist a locally finite refinement that consists of basis sets? The reason this is trickier than it sounds is…
silvascientist
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15
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2 answers

Collection of continuous functions determines the topology

Let $(X,\tau)$ be a topological space. Do the set of all $X\rightarrow X$ continuous functions uniquely determines $\tau$?
Shay Moran
  • 421
15
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8 answers

Why must a topology on a set contain the empty set?

I have just had my first week of topology, and I have a question that is rather basic. Why must the empty set be an element of any given topology? (For reference, the definition of a topology T I am working with, for a set X: 1. X and the empty set…
Cici Shannon
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Axiomatization of a Topology using the Boundary

The axioms of a topological space are usually stated in the "open set" form: A topological space $X$ has a set of subsets $\tau$ whose members satisfy: $\emptyset$ and $X$ are in $\tau$. $\tau$ is closed under arbitrary unions. $\tau$ is closed…
EuYu
  • 41,421
15
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4 answers

An empty intersection of decreasing sequence of closed sets

What is an example of a family of closed subsets $F_0 \supset F_1 \supset F_2 \supset \dots $ of $\mathbb{R}$ so that $F_n \neq \emptyset$ for each $n$ and $\bigcap_{i=1}^n F_i = \emptyset$? Thanks!
user26069
14
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2 answers

Question about a basis for a topology vs the topology generated by a basis?

This is a really basic (no pun intended......no? Ok...) question about what it means to be a basis for a topology. Here is what I know: If $(X, \mathcal{T})$ is a topological space, and $\mathcal{B} \subseteq \mathcal{T}$ is a basis for…
layman
  • 20,191
14
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3 answers

Quotient topologies and equivalence classes

I'm currently studying the notion of a quotient topology. The one thing I'm having trouble with understanding is what we're actually doing to the points as we're identifying them. Say we have a $[0,1] \times [0,1]$ in $E^2$ (euclidean $2$-space)…
Dedalus
  • 3,940
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a continuous bjiective map which is not a homeomorphism

Is there a bijective continuous function $f:\mathbb{Q}\rightarrow \mathbb{Q}$ that not a homeomorphism? I am not able to prove it or disprove it. The problem that the rationals is not even locally compact.
Sobhi
  • 171
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Are simply connected open sets in $\mathbb{R}^2$ homeomorphic to an open ball?

Let $U$ be a simply connected open set in $\mathbb{R}^2$. Is it true that $U$ is homeomorphic to an open ball?
J-Holomorph
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