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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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Quotient map from $\mathbb R^2$ to $\mathbb R$ with cofinite topology

Let $X = \mathbb R$ under the cofinite topology. Is there a quotient map $q : \mathbb R^2 \rightarrow X$? Intuitively, this seems like it should be false, since $\mathbb R^2$ has "too many" open sets. However, I am not sure how to prove it. Any…
user15464
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Line segments intersecting Jordan curve

I have thought this problem a week without success. Is there a set $A\subset \mathbb{R}^2$ such that The boundary of $A$, $\partial A$, is a Jordan curve and For any $B\in \operatorname{int} A\ne\emptyset $, $C\in \operatorname{ext} A\ne\emptyset$…
Jaakko Seppälä
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If $\overline{A}=\overline{B}$, is it true that $\overline{f(A)}=\overline{f(B)}$?

For a continuous function $f:X\longrightarrow Y$ of topological spaces with subsets $A$ and $B$ of $X$ satisfying $\overline{A}=\overline{B}$, is it true that $\overline{f(A)}=\overline{f(B)}$? Intuitively it seems so, but if someone could walk me…
Mathmo
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Definition of uniform structure

From Wikipedia: A uniform space $(X, Φ) $is a set $X$ equipped with a nonempty family $Φ$ of subsets of the Cartesian product $X × X$ ($Φ$ is called the uniform structure or uniformity of $X$ and its elements entourages) that satisfies the…
Tim
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Does continuous imply continuous inverse?

Possible Duplicate: Functions which are Continuous, but not Bicontinuous If $f$ is a continuous map from a subset of $\mathbb{R}^n$ to another subset of $\mathbb{R}^n$, must it have a continuous inverse? (in usual topology) Is the same true of…
veryveryverycoolusername
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limit point of the set of integers

What are the limit point of the set of integers? I know that the closure of this set is the set of integers, so is this set of limit points empty?
user120768
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pre-compactness, total boundedness and "Cauchy sequential compactness"

From Wikipedia: A metric space is called pre-compact or totally bounded if any sequence has a Cauchy subsequence; this can be generalised to uniform spaces. Alternatively, pre-compactness and total boundedness can be defined differently for a…
Tim
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Proving a function is an open map

Let $U$ be an open sub-set of $\mathbb{R}^2$, and let $f$ be a continuous function $f:U\rightarrow \mathbb{R}^2$. I'd like to show that $f$ is an open map, given that for each $u\in U$ exists an open set $V_u$ in $U$ ($V_u\subseteq U$) such that…
Eric_
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The real line, and finite complement topology is not a hausdorff

I am reading through Munkres pg 99 and it says without proof, "the real line in the finite complement topology is nota hausdorff space" i am having trouble getting started with proving this claim .. thank you for a push in the right direction. I…
sophie-germain
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Closure of the difference of two sets vs difference of their closures

This exercise is from Chapter 2, Section 17, number 8 (c), pag. 101, from Munkres's Topology. Let $A$, $B$ denote subsets of a space $X$. Determine whether the following equation holds: $$\overline {A - B} = \overline A - \overline B.$$ If it…
user168731
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Show that if $x$ is a limit point of $A\subset X$ and $f:X \rightarrow Y$ is continuous then $f(x)$ is a limit point of $f(A)$

Suppose $f:X\rightarrow Y$ is a continuous function between the topological spaces $X$ and $Y$. Suppose that: $A \subset X$ and that $x$ is a limit point of $A$. Show that $f(x)$ is a limit point of $f(A)$ By a limit point $c$ of $B$ I mean that…
FNH
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Continuity of the extension of a function between two locally compact Hausdorff spaces to their one point compactifications

Let $X$ and $Y$ be two locally compact Hausdorff spaces, and let $X^+$ and $Y^+$ denote the one point compactifications of $X$ and $Y$, respectively. Let $f: X\rightarrow Y$ be a continuous function and let $f^+: X^+ \rightarrow Y^+$ be the…
Student
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Existence of Open Covers

Do sets always have open covers exist? I know they are not always finite, but do infinite ones always exist? I was reading baby rudin and the proofs for non-relative nature for compactness seems to require that. But I couldn't find any explanations…
user165335
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Counterexample to "Urysohn Lemma for Hausdorff Spaces"

Take a Hausdorff topological space $X$, and two distinct points $p,q \in X$. Is there a continuous function $f: X \to \mathbb{R}$ such that $f(p) \neq f(q)$? The answer is probably no, but I don't have a ready-made counterexample. As soon as we…
Phil Tosteson
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Why sphere and torus are not homeomorphic?

I heard that two objects are homeomorphic if one could be deformed into the other by continuous transformation. For example in this link, it is shown a sphere and a torus are not homeomorphic "Proof" Removing a circle from a sphere always splits…
user26143
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