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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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Why are the real numbers with the K-Topology not metrizable?

I will denote the real numbers with the $K$-Topology as $R_{K}$ (If someone doesn't know or remember this topology, read here). I understand that $R_{K}$ is not regular, since the set $K$ cannot be separated from the point $0$, and it is…
Leta Vitore
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Intersection of connected open sets with union $[0,1]^2$ is connected

I'm trying to show that if the union of two open connected sets equals $[0,1]^2$, then their intersection is also connected. My attempt: Let $U,V \subset [0,1]^2$ be open and connected, and suppose $U \cap V$ is disconnected. Then there exist…
Polly Nomial
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Definition of connectedness and it's intuition

We say a topological space $X$ to be connected if it can not be written as disjoint union of two nonempty open subsets. Intuitively connectedness means our topological space is a single piece.I am not able to see how the above definition captures…
user477157
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Does there Always Exist a Finer/Coarser Topology such that the space is compact?

Sorry for the somewhat vague title. My questions are as follows: Suppose $T$ is a quasicompact space. Does there exist a finer topology on $T$ such that it becomes compact? Suppose $T$ is a Hausdorff space. Does there exist a coarser topology on…
Second Wind
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Inverse image of compact is compact

Let $f : X\to Y$ be a closed map of topological spaces, such that the inverse image of each point in $Y$ is a compact subset of $X$. Is it true that the pre-image of a compact set $K$ is compact? The answer is yes but I’m not sure how to show it. I…
Jama
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"Real Line and Complex plane are the same set just with different topologies" explanation

I was reading the "topology" article on Wikipedia and They stated the following: "For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies." My question is, which set and what…
Eduardo Magalhães
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Properties of the countable complement topology on $\mathbb{R}$.

Let $(\mathbb{R}, \tau)$ be the countable complement topology on the real numbers. I want to know if: (i) $(\mathbb{R}, \tau)$ is metrizable, and if (ii) $(\mathbb{R}, \tau)$ is compact. I think I have (ii) completed. Let…
emka
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How do you show that $\mathbb{Z}^2$ is a closed set in $\mathbb{R}^2$?

In $(\mathbb{R}, \tau_{st})$, we can write $\mathbb{R} \setminus \mathbb{Z} = \bigcup_{n \in \mathbb{Z}} (n,n+1)$, and hence $\mathbb{R} \setminus \mathbb{Z}$ is an open set. Thus the complement, $\mathbb{Z}$, is closed. In $(\mathbb{R}^2,…
masiewpao
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Show that any finite $T_1$-space $X$ is discrete.

Show that any finite $T_1$-space $X$ is discrete. My thoughts:- Let $X=${$a_1 , a_2, …,a_n$} and $U$ be any subset of $X$. Let $U=${$a_1 , a_2, …,a_k$} where$k \le n$ thenwe need to show that $U$ is open that is $X \setminus U$ is closed. Now $X…
poton
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The product of continuous function is continuous.

Statement Let be $A$, $B$, $C$ and $D$ topological spaces and let be $\phi:A\rightarrow C$ and $\psi:B\rightarrow D$ two continuous function. So the product function $\Delta:A\times B\rightarrow C\times D$ defined through the…
Antonio Maria Di Mauro
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A homeomorphism $h$ of the plane $\mathbb{R}^2$ onto itself is given by the formula $h(x, y) = (2x, \frac{1}{2}y).$

I am studying for a qualifying exam. I have been able to do a large percentage of them, but this one in particular has me completely stumped. Could someone give me a start maybe? A homeomorphism $h$ of the plane $\mathbb{R}^2$ onto itself is given…
Caroline
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Show that every uncountable and closed subset of the complete separable metric space contains a homeomorphic subset with the Cantor set.

Show that each uncountable and closed subset of the complete separable metric space contains a homeomorphic subset with the Cantor set. How to show it?
olikalady
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What is the significance of the term "separable" in the context of countability properties?

In the context of topological spaces, I see the following major countability properties: A space is: "separable" iff it has a countable dense subset "second countable" iff if has a countable basis "first countable" iff the neighbourhood system of…
Prime Mover
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Name for an intersection of open subset and closed subset

Is there an established name for a subset of the form $U \cap V$ where $U \subset X$ is open and $V \subset X$ is closed? For example, locally compact subspaces of a locally compact Hausdorff space are exactly of this kind. If there are no existing…
kaba
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Why is the Hawaiian Earring closed?

The Hawaiian Earring $X$ is the union of the circles $[x-(1/n)]^2+y^2=(1/n)^2,n=1,2,3...$ with the topology from the plane. I want to show that $X$ is closed. I note that $X$ is a countable union of closed sets, which is not necessarily closed.…
Roun
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